_divideComplex, real part

Percentage Accurate: 61.7% → 84.1%

Time: 8.8s

Alternatives: 12

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: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+306)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (/ (/ x.im (hypot y.re y.im)) (/ (hypot 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 ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+306) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / hypot(y_46_re, y_46_im)) / (hypot(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 (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+306)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) / Float64(hypot(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_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+306], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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))) < 2.00000000000000003e306

   1. Initial program 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. add-sqr-sqrt 81.0%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. fma-def 81.1%

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

      6. hypot-def 94.7%

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

   3. Applied egg-rr 94.7%

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

   if 2.00000000000000003e306 < (/.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 5.6%

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

   2. Taylor expanded in x.re around 0 4.4%

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

      1. add-sqr-sqrt 4.4%

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

      2. sqrt-div 4.1%

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

      3. hypot-udef 4.1%

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

      4. sqrt-div 4.1%

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

      5. hypot-udef 5.3%

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

      6. times-frac 4.1%

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

      7. add-sqr-sqrt 4.4%

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

      8. times-frac 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. *-commutative 69.6%

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

      2. clear-num 69.7%

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

      3. un-div-inv 69.7%

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

   6. Applied egg-rr 69.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \end{array} \]

Alternative 2: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -5.9 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (/ x.im (hypot y.re y.im)) (/ y.im (hypot y.re y.im)))))
   (if (<= y.im -5.2e+41)
     t_1
     (if (<= y.im -5.9e-130)
       t_0
       (if (<= y.im 3.8e-142)
         (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
         (if (<= y.im 3.05e+24) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / hypot(y_46_re, y_46_im)) * (y_46_im / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -5.2e+41) {
		tmp = t_1;
	} else if (y_46_im <= -5.9e-130) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-142) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 3.05e+24) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / Math.hypot(y_46_re, y_46_im)) * (y_46_im / Math.hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -5.2e+41) {
		tmp = t_1;
	} else if (y_46_im <= -5.9e-130) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-142) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 3.05e+24) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / math.hypot(y_46_re, y_46_im)) * (y_46_im / math.hypot(y_46_re, y_46_im))
	tmp = 0
	if y_46_im <= -5.2e+41:
		tmp = t_1
	elif y_46_im <= -5.9e-130:
		tmp = t_0
	elif y_46_im <= 3.8e-142:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_im <= 3.05e+24:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) * Float64(y_46_im / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_im <= -5.2e+41)
		tmp = t_1;
	elseif (y_46_im <= -5.9e-130)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-142)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_im <= 3.05e+24)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / hypot(y_46_re, y_46_im)) * (y_46_im / hypot(y_46_re, y_46_im));
	tmp = 0.0;
	if (y_46_im <= -5.2e+41)
		tmp = t_1;
	elseif (y_46_im <= -5.9e-130)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-142)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_im <= 3.05e+24)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.2e+41], t$95$1, If[LessEqual[y$46$im, -5.9e-130], t$95$0, If[LessEqual[y$46$im, 3.8e-142], 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], If[LessEqual[y$46$im, 3.05e+24], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.2000000000000001e41 or 3.05000000000000003e24 < y.im

   1. Initial program 45.3%

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

   2. Taylor expanded in x.re around 0 42.4%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-div 20.5%

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

      3. hypot-udef 20.5%

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

      4. sqrt-div 20.5%

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

      5. hypot-udef 26.0%

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

      6. times-frac 20.5%

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

      7. add-sqr-sqrt 42.4%

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

      8. times-frac 84.7%

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

   4. Applied egg-rr 84.7%

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

   if -5.2000000000000001e41 < y.im < -5.9000000000000003e-130 or 3.79999999999999972e-142 < y.im < 3.05000000000000003e24

   1. Initial program 90.1%

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

   if -5.9000000000000003e-130 < y.im < 3.79999999999999972e-142

   1. Initial program 73.8%

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

   2. Taylor expanded in y.re around inf 87.7%

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

      1. unpow2 87.7%

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

      2. times-frac 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -5.9 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (/ x.im (hypot y.re y.im)) (/ (hypot y.re y.im) y.im))))
   (if (<= y.im -7.5e+48)
     t_1
     (if (<= y.im -3.05e-130)
       t_0
       (if (<= y.im 1.55e-142)
         (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
         (if (<= y.im 1.4e+25) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / hypot(y_46_re, y_46_im)) / (hypot(y_46_re, y_46_im) / y_46_im);
	double tmp;
	if (y_46_im <= -7.5e+48) {
		tmp = t_1;
	} else if (y_46_im <= -3.05e-130) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e-142) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 1.4e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / Math.hypot(y_46_re, y_46_im)) / (Math.hypot(y_46_re, y_46_im) / y_46_im);
	double tmp;
	if (y_46_im <= -7.5e+48) {
		tmp = t_1;
	} else if (y_46_im <= -3.05e-130) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e-142) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 1.4e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / math.hypot(y_46_re, y_46_im)) / (math.hypot(y_46_re, y_46_im) / y_46_im)
	tmp = 0
	if y_46_im <= -7.5e+48:
		tmp = t_1
	elif y_46_im <= -3.05e-130:
		tmp = t_0
	elif y_46_im <= 1.55e-142:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_im <= 1.4e+25:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) / Float64(hypot(y_46_re, y_46_im) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -7.5e+48)
		tmp = t_1;
	elseif (y_46_im <= -3.05e-130)
		tmp = t_0;
	elseif (y_46_im <= 1.55e-142)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_im <= 1.4e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / hypot(y_46_re, y_46_im)) / (hypot(y_46_re, y_46_im) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -7.5e+48)
		tmp = t_1;
	elseif (y_46_im <= -3.05e-130)
		tmp = t_0;
	elseif (y_46_im <= 1.55e-142)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_im <= 1.4e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+48], t$95$1, If[LessEqual[y$46$im, -3.05e-130], t$95$0, If[LessEqual[y$46$im, 1.55e-142], 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], If[LessEqual[y$46$im, 1.4e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.5000000000000006e48 or 1.4000000000000001e25 < y.im

   1. Initial program 45.3%

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

   2. Taylor expanded in x.re around 0 42.4%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-div 20.5%

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

      3. hypot-udef 20.5%

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

      4. sqrt-div 20.5%

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

      5. hypot-udef 26.0%

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

      6. times-frac 20.5%

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

      7. add-sqr-sqrt 42.4%

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

      8. times-frac 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. clear-num 84.7%

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

      3. un-div-inv 84.8%

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

   6. Applied egg-rr 84.8%

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

   if -7.5000000000000006e48 < y.im < -3.04999999999999998e-130 or 1.55e-142 < y.im < 1.4000000000000001e25

   1. Initial program 90.1%

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

   if -3.04999999999999998e-130 < y.im < 1.55e-142

   1. Initial program 73.8%

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

   2. Taylor expanded in y.re around inf 87.7%

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

      1. unpow2 87.7%

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

      2. times-frac 88.6%

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

   4. Simplified 88.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \end{array} \]

Alternative 4: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \end{array} \]

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 (<= y.im -1.15e+73)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.im -3.2e-129)
       t_0
       (if (<= y.im 3.7e-143)
         (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
         (if (<= y.im 6e+66)
           t_0
           (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.15e+73) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_im <= -3.2e-129) {
		tmp = t_0;
	} else if (y_46_im <= 3.7e-143) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 6e+66) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.15d+73)) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46im <= (-3.2d-129)) then
        tmp = t_0
    else if (y_46im <= 3.7d-143) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else if (y_46im <= 6d+66) then
        tmp = t_0
    else
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (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 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 (y_46_im <= -1.15e+73) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_im <= -3.2e-129) {
		tmp = t_0;
	} else if (y_46_im <= 3.7e-143) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= 6e+66) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.15e+73:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_im <= -3.2e-129:
		tmp = t_0
	elif y_46_im <= 3.7e-143:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_im <= 6e+66:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.15e+73)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_im <= -3.2e-129)
		tmp = t_0;
	elseif (y_46_im <= 3.7e-143)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_im <= 6e+66)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * 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)
	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 (y_46_im <= -1.15e+73)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_im <= -3.2e-129)
		tmp = t_0;
	elseif (y_46_im <= 3.7e-143)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_im <= 6e+66)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (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_] := 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[y$46$im, -1.15e+73], 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], If[LessEqual[y$46$im, -3.2e-129], t$95$0, If[LessEqual[y$46$im, 3.7e-143], 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], If[LessEqual[y$46$im, 6e+66], t$95$0, 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.15e73

   1. Initial program 46.7%

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

   2. Taylor expanded in y.re around 0 74.1%

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

      1. +-commutative 74.1%

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

      2. *-commutative 74.1%

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

      3. unpow2 74.1%

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

      4. times-frac 84.6%

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

   4. Simplified 84.6%

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

   if -1.15e73 < y.im < -3.2000000000000003e-129 or 3.7e-143 < y.im < 6.00000000000000005e66

   1. Initial program 86.1%

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

   if -3.2000000000000003e-129 < y.im < 3.7e-143

   1. Initial program 73.8%

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

   2. Taylor expanded in y.re around inf 87.7%

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

      1. unpow2 87.7%

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

      2. times-frac 88.6%

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

   4. Simplified 88.6%

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

   if 6.00000000000000005e66 < y.im

   1. Initial program 41.1%

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

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

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

      1. +-commutative 76.4%

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

      2. *-commutative 76.4%

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

      3. unpow2 76.4%

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

      4. associate-/l* 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in y.im around 0 78.5%

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

      1. unpow2 78.5%

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

      2. associate-*r/ 80.9%

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

   7. Simplified 80.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 5: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.re -6e-13)
     (/ x.re y.re)
     (if (<= y.re 1.45e-30)
       t_0
       (if (<= y.re 3.4e-6)
         (/ y.im (/ (* y.re y.re) x.im))
         (if (<= y.re 6.2e+15) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -6e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 6.2e+15) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46re <= (-6d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.45d-30) then
        tmp = t_0
    else if (y_46re <= 3.4d-6) then
        tmp = y_46im / ((y_46re * y_46re) / x_46im)
    else if (y_46re <= 6.2d+15) then
        tmp = t_0
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -6e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 6.2e+15) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -6e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.45e-30:
		tmp = t_0
	elif y_46_re <= 3.4e-6:
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im)
	elif y_46_re <= 6.2e+15:
		tmp = t_0
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.45e-30)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-6)
		tmp = Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_im));
	elseif (y_46_re <= 6.2e+15)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.45e-30)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-6)
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	elseif (y_46_re <= 6.2e+15)
		tmp = t_0;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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]}, If[LessEqual[y$46$re, -6e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-30], t$95$0, If[LessEqual[y$46$re, 3.4e-6], N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+15], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.99999999999999968e-13 or 6.2e15 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -5.99999999999999968e-13 < y.re < 1.44999999999999995e-30 or 3.40000000000000006e-6 < y.re < 6.2e15

   1. Initial program 72.5%

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

   2. Taylor expanded in y.re around 0 82.7%

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

      1. +-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. unpow2 82.7%

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

      4. times-frac 86.2%

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

   4. Simplified 86.2%

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

   if 1.44999999999999995e-30 < y.re < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in x.re around 0 83.5%

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

   3. Taylor expanded in y.im around 0 83.5%

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

      1. associate-/l* 83.8%

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

      2. unpow2 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 6: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.2e-13)
   (/ x.re y.re)
   (if (<= y.re 1.45e-30)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.re 3.4e-6)
       (/ y.im (/ (* y.re y.re) x.im))
       (if (<= y.re 3.5e+14)
         (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
         (/ 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_re <= -3.2e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 3.5e+14) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} 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_46re <= (-3.2d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.45d-30) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 3.4d-6) then
        tmp = y_46im / ((y_46re * y_46re) / x_46im)
    else if (y_46re <= 3.5d+14) then
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (y_46im / x_46re)))
    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_re <= -3.2e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 3.5e+14) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} 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_re <= -3.2e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.45e-30:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 3.4e-6:
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im)
	elif y_46_re <= 3.5e+14:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	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_re <= -3.2e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.45e-30)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 3.4e-6)
		tmp = Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_im));
	elseif (y_46_re <= 3.5e+14)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	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_re <= -3.2e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.45e-30)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 3.4e-6)
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	elseif (y_46_re <= 3.5e+14)
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	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[LessEqual[y$46$re, -3.2e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-30], 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], If[LessEqual[y$46$re, 3.4e-6], N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+14], 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], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.2e-13 or 3.5e14 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -3.2e-13 < y.re < 1.44999999999999995e-30

   1. Initial program 72.9%

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

   2. Taylor expanded in y.re around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 86.4%

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

   4. Simplified 86.4%

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

   if 1.44999999999999995e-30 < y.re < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in x.re around 0 83.5%

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

   3. Taylor expanded in y.im around 0 83.5%

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

      1. associate-/l* 83.8%

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

      2. unpow2 83.8%

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

   5. Simplified 83.8%

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

   if 3.40000000000000006e-6 < y.re < 3.5e14

   1. Initial program 60.8%

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

   2. Taylor expanded in y.re around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. *-commutative 80.4%

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

      3. unpow2 80.4%

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

      4. associate-/l* 80.2%

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

   4. Simplified 80.2%

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

   5. Taylor expanded in y.im around 0 80.2%

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

      1. unpow2 80.2%

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

      2. associate-*r/ 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-13} \lor \neg \left(y.re \leq 1.45 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e-13) (not (<= y.re 1.45e-30)))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im 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_re <= -6e-13) || !(y_46_re <= 1.45e-30)) {
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6d-13)) .or. (.not. (y_46re <= 1.45d-30))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / 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_re <= -6e-13) || !(y_46_re <= 1.45e-30)) {
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6e-13) or not (y_46_re <= 1.45e-30):
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6e-13) || !(y_46_re <= 1.45e-30))
		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(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_re <= -6e-13) || ~((y_46_re <= 1.45e-30)))
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6e-13], N[Not[LessEqual[y$46$re, 1.45e-30]], $MachinePrecision]], 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[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.99999999999999968e-13 or 1.44999999999999995e-30 < y.re

   1. Initial program 55.6%

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

   2. Taylor expanded in y.re around inf 70.7%

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

      1. unpow2 70.7%

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

      2. times-frac 78.1%

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

   4. Simplified 78.1%

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

   if -5.99999999999999968e-13 < y.re < 1.44999999999999995e-30

   1. Initial program 72.9%

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

   2. Taylor expanded in y.re around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. *-commutative 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-13} \lor \neg \left(y.re \leq 1.45 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 17000000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e-13)
   (/ x.re y.re)
   (if (<= y.re 1.45e-30)
     (/ x.im y.im)
     (if (<= y.re 4.1e-6)
       (* y.im (/ x.im (* y.re y.re)))
       (if (<= y.re 17000000000000.0) (/ x.im y.im) (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 4.1e-6) {
		tmp = y_46_im * (x_46_im / (y_46_re * y_46_re));
	} else if (y_46_re <= 17000000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.2d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.45d-30) then
        tmp = x_46im / y_46im
    else if (y_46re <= 4.1d-6) then
        tmp = y_46im * (x_46im / (y_46re * y_46re))
    else if (y_46re <= 17000000000000.0d0) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 4.1e-6) {
		tmp = y_46_im * (x_46_im / (y_46_re * y_46_re));
	} else if (y_46_re <= 17000000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.2e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.45e-30:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 4.1e-6:
		tmp = y_46_im * (x_46_im / (y_46_re * y_46_re))
	elif y_46_re <= 17000000000000.0:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.45e-30)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 4.1e-6)
		tmp = Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 17000000000000.0)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.2e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.45e-30)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 4.1e-6)
		tmp = y_46_im * (x_46_im / (y_46_re * y_46_re));
	elseif (y_46_re <= 17000000000000.0)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.2e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-30], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.1e-6], N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 17000000000000.0], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.19999999999999977e-13 or 1.7e13 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -4.19999999999999977e-13 < y.re < 1.44999999999999995e-30 or 4.0999999999999997e-6 < y.re < 1.7e13

   1. Initial program 72.5%

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

   2. Taylor expanded in y.re around 0 74.0%

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

   if 1.44999999999999995e-30 < y.re < 4.0999999999999997e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in x.re around 0 83.5%

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

   3. Taylor expanded in y.im around 0 83.5%

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

      1. associate-/l* 83.8%

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

      2. unpow2 83.8%

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

   5. Simplified 83.8%

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

      1. div-inv 83.0%

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

      2. clear-num 83.2%

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

   7. Applied egg-rr 83.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{-6}:\\ \;\;\;\;y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 17000000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{y.re \cdot \frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 20000000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e-13)
   (/ x.re y.re)
   (if (<= y.re 1.45e-30)
     (/ x.im y.im)
     (if (<= y.re 3.4e-6)
       (/ y.im (* y.re (/ y.re x.im)))
       (if (<= y.re 20000000000000.0) (/ x.im y.im) (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / (y_46_re * (y_46_re / x_46_im));
	} else if (y_46_re <= 20000000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.4d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.45d-30) then
        tmp = x_46im / y_46im
    else if (y_46re <= 3.4d-6) then
        tmp = y_46im / (y_46re * (y_46re / x_46im))
    else if (y_46re <= 20000000000000.0d0) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.4e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / (y_46_re * (y_46_re / x_46_im));
	} else if (y_46_re <= 20000000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.4e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.45e-30:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 3.4e-6:
		tmp = y_46_im / (y_46_re * (y_46_re / x_46_im))
	elif y_46_re <= 20000000000000.0:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.4e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.45e-30)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 3.4e-6)
		tmp = Float64(y_46_im / Float64(y_46_re * Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 20000000000000.0)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.4e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.45e-30)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 3.4e-6)
		tmp = y_46_im / (y_46_re * (y_46_re / x_46_im));
	elseif (y_46_re <= 20000000000000.0)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.4e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-30], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.4e-6], N[(y$46$im / N[(y$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 20000000000000.0], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.3999999999999999e-13 or 2e13 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -2.3999999999999999e-13 < y.re < 1.44999999999999995e-30 or 3.40000000000000006e-6 < y.re < 2e13

   1. Initial program 72.5%

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

   2. Taylor expanded in y.re around 0 74.0%

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

   if 1.44999999999999995e-30 < y.re < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in x.re around 0 83.5%

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

   3. Taylor expanded in y.im around 0 83.5%

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

      1. associate-/l* 83.8%

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

      2. unpow2 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in y.re around 0 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 83.4%

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

   8. Simplified 83.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{y.re \cdot \frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 20000000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 4900000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6e-13)
   (/ x.re y.re)
   (if (<= y.re 1.45e-30)
     (/ x.im y.im)
     (if (<= y.re 3.4e-6)
       (/ y.im (/ (* y.re y.re) x.im))
       (if (<= y.re 4900000000000.0) (/ x.im y.im) (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 4900000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-6d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.45d-30) then
        tmp = x_46im / y_46im
    else if (y_46re <= 3.4d-6) then
        tmp = y_46im / ((y_46re * y_46re) / x_46im)
    else if (y_46re <= 4900000000000.0d0) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.45e-30) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 3.4e-6) {
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	} else if (y_46_re <= 4900000000000.0) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -6e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.45e-30:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 3.4e-6:
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im)
	elif y_46_re <= 4900000000000.0:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -6e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.45e-30)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 3.4e-6)
		tmp = Float64(y_46_im / Float64(Float64(y_46_re * y_46_re) / x_46_im));
	elseif (y_46_re <= 4900000000000.0)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -6e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.45e-30)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 3.4e-6)
		tmp = y_46_im / ((y_46_re * y_46_re) / x_46_im);
	elseif (y_46_re <= 4900000000000.0)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-30], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.4e-6], N[(y$46$im / N[(N[(y$46$re * y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4900000000000.0], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.99999999999999968e-13 or 4.9e12 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -5.99999999999999968e-13 < y.re < 1.44999999999999995e-30 or 3.40000000000000006e-6 < y.re < 4.9e12

   1. Initial program 72.5%

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

   2. Taylor expanded in y.re around 0 74.0%

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

   if 1.44999999999999995e-30 < y.re < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in x.re around 0 83.5%

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

   3. Taylor expanded in y.im around 0 83.5%

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

      1. associate-/l* 83.8%

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

      2. unpow2 83.8%

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

   5. Simplified 83.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 4900000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.5e-13)
   (/ x.re y.re)
   (if (<= y.re 3.1e+14) (/ x.im y.im) (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.5e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+14) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-5.5d-13)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.1d+14) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.5e-13) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+14) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5.5e-13:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.1e+14:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.5e-13)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.1e+14)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5.5e-13)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.1e+14)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.5e-13], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.1e+14], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.49999999999999979e-13 or 3.1e14 < y.re

   1. Initial program 53.3%

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

   2. Taylor expanded in y.re around inf 66.2%

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

   if -5.49999999999999979e-13 < y.re < 3.1e14

   1. Initial program 73.7%

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

   2. Taylor expanded in y.re around 0 70.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12: 43.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.9%

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

2. Taylor expanded in y.re around 0 47.2%

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

3. Final simplification 47.2%

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

Reproduce

herbie shell --seed 2023182 
(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))))