_divideComplex, imaginary part

Percentage Accurate: 61.1% → 84.7%

Time: 16.8s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+69}:\\ \;\;\;\;t_0 \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (* t_0 (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im)))))
   (if (<= y.re -9e+69)
     (* t_0 (- (* y.im (/ x.re y.re)) x.im))
     (if (<= y.re -5.5e-101)
       t_1
       (if (<= y.re 1.1e-227)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 2.5e+81)
           t_1
           (/ (- x.im (* x.re (/ y.im y.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 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -9e+69) {
		tmp = t_0 * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -5.5e-101) {
		tmp = t_1;
	} else if (y_46_re <= 1.1e-227) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.5e+81) {
		tmp = t_1;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -9e+69) {
		tmp = t_0 * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -5.5e-101) {
		tmp = t_1;
	} else if (y_46_re <= 1.1e-227) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.5e+81) {
		tmp = t_1;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	tmp = 0
	if y_46_re <= -9e+69:
		tmp = t_0 * ((y_46_im * (x_46_re / y_46_re)) - x_46_im)
	elif y_46_re <= -5.5e-101:
		tmp = t_1
	elif y_46_re <= 1.1e-227:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 2.5e+81:
		tmp = t_1
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(t_0 * Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9e+69)
		tmp = Float64(t_0 * Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im));
	elseif (y_46_re <= -5.5e-101)
		tmp = t_1;
	elseif (y_46_re <= 1.1e-227)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.5e+81)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_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 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	tmp = 0.0;
	if (y_46_re <= -9e+69)
		tmp = t_0 * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	elseif (y_46_re <= -5.5e-101)
		tmp = t_1;
	elseif (y_46_re <= 1.1e-227)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 2.5e+81)
		tmp = t_1;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+69], N[(t$95$0 * N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.5e-101], t$95$1, If[LessEqual[y$46$re, 1.1e-227], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+81], t$95$1, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.9999999999999999e69

   1. Initial program 40.2%

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

      1. *-un-lft-identity 40.2%

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

      2. add-sqr-sqrt 40.2%

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

      3. times-frac 40.2%

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

      4. hypot-def 40.2%

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

      5. hypot-def 59.5%

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

   3. Applied egg-rr 59.5%

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

   4. Taylor expanded in y.re around -inf 76.1%

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

      1. +-commutative 76.1%

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

      2. mul-1-neg 76.1%

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

      3. unsub-neg 76.1%

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

      4. associate-/l* 82.4%

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

      5. associate-/r/ 82.3%

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

   6. Simplified 82.3%

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

   if -8.9999999999999999e69 < y.re < -5.49999999999999973e-101 or 1.0999999999999999e-227 < y.re < 2.4999999999999999e81

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 85.7%

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

      4. hypot-def 85.7%

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

      5. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

   if -5.49999999999999973e-101 < y.re < 1.0999999999999999e-227

   1. Initial program 66.9%

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

   2. Taylor expanded in y.re around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. *-commutative 87.8%

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

      5. unpow2 87.8%

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

      6. times-frac 91.2%

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

   4. Simplified 91.2%

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

      1. associate-*r/ 95.6%

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

      2. sub-div 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 2.4999999999999999e81 < y.re

   1. Initial program 36.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. unpow2 81.7%

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

      5. associate-/l* 84.0%

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

      6. associate-/r/ 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.re around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x.im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. *-commutative 81.7%

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

      4. unpow2 81.7%

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

      5. times-frac 92.1%

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

      6. distribute-lft-neg-in 92.1%

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

      7. cancel-sign-sub-inv 92.1%

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

      8. associate-*r/ 92.1%

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

      9. *-commutative 92.1%

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

      10. div-sub 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 2: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;t_0 \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= y.re -4.2e+52)
     (* (/ 1.0 (hypot y.re y.im)) (- (* y.im (/ x.re y.re)) x.im))
     (if (<= y.re -5.5e-106)
       (* t_0 (/ 1.0 (pow (hypot y.re y.im) 2.0)))
       (if (<= y.re 1.45e-160)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 5.5e+80)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -4.2e+52) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -5.5e-106) {
		tmp = t_0 * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else if (y_46_re <= 1.45e-160) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 5.5e+80) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -4.2e+52) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -5.5e-106) {
		tmp = t_0 * (1.0 / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0));
	} else if (y_46_re <= 1.45e-160) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 5.5e+80) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if y_46_re <= -4.2e+52:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im)
	elif y_46_re <= -5.5e-106:
		tmp = t_0 * (1.0 / math.pow(math.hypot(y_46_re, y_46_im), 2.0))
	elif y_46_re <= 1.45e-160:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 5.5e+80:
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.2e+52)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im));
	elseif (y_46_re <= -5.5e-106)
		tmp = Float64(t_0 * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	elseif (y_46_re <= 1.45e-160)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 5.5e+80)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.2e+52)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	elseif (y_46_re <= -5.5e-106)
		tmp = t_0 * (1.0 / (hypot(y_46_re, y_46_im) ^ 2.0));
	elseif (y_46_re <= 1.45e-160)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 5.5e+80)
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.2e+52], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.5e-106], N[(t$95$0 * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-160], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.5e+80], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.2e52

   1. Initial program 44.7%

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

      1. *-un-lft-identity 44.7%

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

      2. add-sqr-sqrt 44.7%

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

      3. times-frac 44.8%

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

      4. hypot-def 44.8%

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

      5. hypot-def 62.6%

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

   3. Applied egg-rr 62.6%

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

   4. Taylor expanded in y.re around -inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. associate-/l* 83.7%

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

      5. associate-/r/ 83.6%

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

   6. Simplified 83.6%

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

   if -4.2e52 < y.re < -5.5000000000000001e-106

   1. Initial program 88.6%

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

      1. clear-num 88.4%

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

      2. associate-/r/ 88.7%

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

      3. add-sqr-sqrt 88.6%

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

      4. pow2 88.6%

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

      5. hypot-def 88.7%

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

   3. Applied egg-rr 88.7%

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

   if -5.5000000000000001e-106 < y.re < 1.45e-160

   1. Initial program 68.4%

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

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

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 1.45e-160 < y.re < 5.49999999999999967e80

   1. Initial program 86.6%

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

   if 5.49999999999999967e80 < y.re

   1. Initial program 36.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. unpow2 81.7%

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

      5. associate-/l* 84.0%

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

      6. associate-/r/ 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.re around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x.im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. *-commutative 81.7%

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

      4. unpow2 81.7%

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

      5. times-frac 92.1%

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

      6. distribute-lft-neg-in 92.1%

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

      7. cancel-sign-sub-inv 92.1%

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

      8. associate-*r/ 92.1%

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

      9. *-commutative 92.1%

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

      10. div-sub 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-106}:\\ \;\;\;\;\left(y.re \cdot x.im - y.im \cdot x.re\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 3: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= y.re -5.2e+57)
     (* (/ 1.0 (hypot y.re y.im)) (- (* y.im (/ x.re y.re)) x.im))
     (if (<= y.re -2.7e-108)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.re 1.45e-160)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 4.4e+79)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -5.2e+57) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -2.7e-108) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 1.45e-160) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 4.4e+79) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_re <= -5.2e+57)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im));
	elseif (y_46_re <= -2.7e-108)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.45e-160)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 4.4e+79)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.2e+57], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.7e-108], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.45e-160], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.4e+79], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -5.2e57

   1. Initial program 43.6%

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

      1. *-un-lft-identity 43.6%

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

      2. add-sqr-sqrt 43.6%

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

      3. times-frac 43.8%

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

      4. hypot-def 43.8%

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

      5. hypot-def 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in y.re around -inf 77.5%

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

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-/l* 83.4%

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

      5. associate-/r/ 83.3%

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

   6. Simplified 83.3%

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

   if -5.2e57 < y.re < -2.70000000000000005e-108

   1. Initial program 89.0%

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

      1. fma-def 89.0%

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

   3. Simplified 89.0%

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

   if -2.70000000000000005e-108 < y.re < 1.45e-160

   1. Initial program 68.4%

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

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

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 1.45e-160 < y.re < 4.3999999999999998e79

   1. Initial program 86.6%

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

   if 4.3999999999999998e79 < y.re

   1. Initial program 36.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. unpow2 81.7%

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

      5. associate-/l* 84.0%

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

      6. associate-/r/ 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.re around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x.im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. *-commutative 81.7%

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

      4. unpow2 81.7%

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

      5. times-frac 92.1%

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

      6. distribute-lft-neg-in 92.1%

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

      7. cancel-sign-sub-inv 92.1%

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

      8. associate-*r/ 92.1%

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

      9. *-commutative 92.1%

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

      10. div-sub 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 4: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.35 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.2e+57)
     (* (/ 1.0 (hypot y.re y.im)) (- (* y.im (/ x.re y.re)) x.im))
     (if (<= y.re -1.9e-106)
       t_0
       (if (<= y.re 5.5e-161)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 3.35e+77)
           t_0
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.2e+57) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -1.9e-106) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e-161) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.35e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.2e+57) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	} else if (y_46_re <= -1.9e-106) {
		tmp = t_0;
	} else if (y_46_re <= 5.5e-161) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.35e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.2e+57:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im)
	elif y_46_re <= -1.9e-106:
		tmp = t_0
	elif y_46_re <= 5.5e-161:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 3.35e+77:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.2e+57)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im));
	elseif (y_46_re <= -1.9e-106)
		tmp = t_0;
	elseif (y_46_re <= 5.5e-161)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.35e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.2e+57)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_46_im);
	elseif (y_46_re <= -1.9e-106)
		tmp = t_0;
	elseif (y_46_re <= 5.5e-161)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 3.35e+77)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.2e+57], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.9e-106], t$95$0, If[LessEqual[y$46$re, 5.5e-161], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.35e+77], t$95$0, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.20000000000000002e57

   1. Initial program 43.6%

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

      1. *-un-lft-identity 43.6%

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

      2. add-sqr-sqrt 43.6%

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

      3. times-frac 43.8%

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

      4. hypot-def 43.8%

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

      5. hypot-def 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in y.re around -inf 77.5%

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

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-/l* 83.4%

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

      5. associate-/r/ 83.3%

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

   6. Simplified 83.3%

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

   if -1.20000000000000002e57 < y.re < -1.9e-106 or 5.5e-161 < y.re < 3.35000000000000014e77

   1. Initial program 87.5%

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

   if -1.9e-106 < y.re < 5.5e-161

   1. Initial program 68.4%

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

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

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 3.35000000000000014e77 < y.re

   1. Initial program 36.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. unpow2 81.7%

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

      5. associate-/l* 84.0%

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

      6. associate-/r/ 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.re around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x.im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. *-commutative 81.7%

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

      4. unpow2 81.7%

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

      5. times-frac 92.1%

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

      6. distribute-lft-neg-in 92.1%

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

      7. cancel-sign-sub-inv 92.1%

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

      8. associate-*r/ 92.1%

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

      9. *-commutative 92.1%

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

      10. div-sub 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.35 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 5: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -6.1e+57)
     (+ (/ x.im y.re) (/ -1.0 (* (/ y.re y.im) (/ y.re x.re))))
     (if (<= y.re -1.15e-103)
       t_0
       (if (<= y.re 1.16e-160)
         (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
         (if (<= y.re 2.6e+77)
           t_0
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.1e+57) {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -1.15e-103) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-160) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.6e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-6.1d+57)) then
        tmp = (x_46im / y_46re) + ((-1.0d0) / ((y_46re / y_46im) * (y_46re / x_46re)))
    else if (y_46re <= (-1.15d-103)) then
        tmp = t_0
    else if (y_46re <= 1.16d-160) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46re <= 2.6d+77) then
        tmp = t_0
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.1e+57) {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -1.15e-103) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-160) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.6e+77) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -6.1e+57:
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)))
	elif y_46_re <= -1.15e-103:
		tmp = t_0
	elif y_46_re <= 1.16e-160:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 2.6e+77:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6.1e+57)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(-1.0 / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / x_46_re))));
	elseif (y_46_re <= -1.15e-103)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-160)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.6e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6.1e+57)
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	elseif (y_46_re <= -1.15e-103)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-160)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 2.6e+77)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.1e+57], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(-1.0 / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.15e-103], t$95$0, If[LessEqual[y$46$re, 1.16e-160], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.6e+77], t$95$0, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.09999999999999975e57

   1. Initial program 43.6%

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

   2. Taylor expanded in y.re around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. associate-/l* 70.3%

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

      6. associate-/r/ 74.2%

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

   4. Simplified 74.2%

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

   5. Taylor expanded in x.re around 0 73.5%

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

      1. unpow2 73.5%

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

      2. times-frac 83.2%

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

   7. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. clear-num 83.2%

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

      3. clear-num 83.1%

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

      4. frac-times 83.3%

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

      5. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

   if -6.09999999999999975e57 < y.re < -1.15e-103 or 1.16e-160 < y.re < 2.6000000000000002e77

   1. Initial program 87.5%

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

   if -1.15e-103 < y.re < 1.16e-160

   1. Initial program 68.4%

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

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

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 2.6000000000000002e77 < y.re

   1. Initial program 36.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. unpow2 81.7%

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

      5. associate-/l* 84.0%

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

      6. associate-/r/ 83.9%

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

   4. Simplified 83.9%

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

   5. Taylor expanded in x.re around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 92.1%

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

   7. Simplified 92.1%

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

   8. Taylor expanded in x.im around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. *-commutative 81.7%

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

      4. unpow2 81.7%

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

      5. times-frac 92.1%

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

      6. distribute-lft-neg-in 92.1%

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

      7. cancel-sign-sub-inv 92.1%

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

      8. associate-*r/ 92.1%

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

      9. *-commutative 92.1%

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

      10. div-sub 92.1%

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

   10. Simplified 92.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-28} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.4e-28) (not (<= y.re 8.5e-151)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.4e-28) || !(y_46_re <= 8.5e-151)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.4d-28)) .or. (.not. (y_46re <= 8.5d-151))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.4e-28) || !(y_46_re <= 8.5e-151)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.4e-28) or not (y_46_re <= 8.5e-151):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.4e-28) || !(y_46_re <= 8.5e-151))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.4e-28) || ~((y_46_re <= 8.5e-151)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.4e-28], N[Not[LessEqual[y$46$re, 8.5e-151]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.3999999999999999e-28 or 8.49999999999999999e-151 < y.re

   1. Initial program 58.0%

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

   2. Taylor expanded in y.re around inf 71.4%

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

      1. +-commutative 71.4%

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

      2. mul-1-neg 71.4%

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

      3. unsub-neg 71.4%

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

      4. unpow2 71.4%

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

      5. associate-/l* 69.8%

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

      6. associate-/r/ 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in x.re around 0 71.4%

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

      1. unpow2 71.4%

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

      2. times-frac 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x.im around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. mul-1-neg 71.4%

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

      3. *-commutative 71.4%

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

      4. unpow2 71.4%

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

      5. times-frac 77.2%

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

      6. distribute-lft-neg-in 77.2%

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

      7. cancel-sign-sub-inv 77.2%

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

      8. associate-*r/ 77.2%

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

      9. *-commutative 77.2%

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

      10. div-sub 77.2%

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

   10. Simplified 77.2%

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

   if -1.3999999999999999e-28 < y.re < 8.49999999999999999e-151

   1. Initial program 70.9%

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

   2. Taylor expanded in y.re around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-28} \lor \neg \left(y.re \leq 8.5 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 7: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e-27)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 5.1e-151)
     (/ (- x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.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 <= -2.6e-27) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 5.1e-151) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= (-2.6d-27)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 5.1d-151) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_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 <= -2.6e-27) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 5.1e-151) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -2.6e-27:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 5.1e-151:
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -2.6e-27)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 5.1e-151)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_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 <= -2.6e-27)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 5.1e-151)
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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, -2.6e-27], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 5.1e-151], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.60000000000000017e-27

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in x.re around 0 74.3%

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

      1. unpow2 74.3%

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

      2. times-frac 82.4%

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

   7. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   9. Applied egg-rr 83.7%

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

   if -2.60000000000000017e-27 < y.re < 5.0999999999999997e-151

   1. Initial program 70.9%

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

   2. Taylor expanded in y.re around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

   if 5.0999999999999997e-151 < y.re

   1. Initial program 60.9%

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

   2. Taylor expanded in y.re around inf 69.4%

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

      1. +-commutative 69.4%

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

      2. mul-1-neg 69.4%

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

      3. unsub-neg 69.4%

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

      4. unpow2 69.4%

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

      5. associate-/l* 70.6%

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

      6. associate-/r/ 67.5%

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

   4. Simplified 67.5%

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

   5. Taylor expanded in x.re around 0 69.4%

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

      1. unpow2 69.4%

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

      2. times-frac 73.8%

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

   7. Simplified 73.8%

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

   8. Taylor expanded in x.im around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. mul-1-neg 69.4%

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

      3. *-commutative 69.4%

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

      4. unpow2 69.4%

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

      5. times-frac 73.8%

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

      6. distribute-lft-neg-in 73.8%

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

      7. cancel-sign-sub-inv 73.8%

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

      8. associate-*r/ 74.8%

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

      9. *-commutative 74.8%

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

      10. div-sub 74.8%

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

   10. Simplified 74.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{-151}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.45e-30)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 1.2e-11)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.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 <= -1.45e-30) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.2e-11) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= (-1.45d-30)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 1.2d-11) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_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 <= -1.45e-30) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.2e-11) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -1.45e-30:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 1.2e-11:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -1.45e-30)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.2e-11)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_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 <= -1.45e-30)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.2e-11)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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, -1.45e-30], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.2e-11], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.44999999999999995e-30

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in x.re around 0 74.3%

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

      1. unpow2 74.3%

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

      2. times-frac 82.4%

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

   7. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   9. Applied egg-rr 83.7%

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

   if -1.44999999999999995e-30 < y.re < 1.2000000000000001e-11

   1. Initial program 75.2%

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

   2. Taylor expanded in y.re around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

      6. times-frac 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in y.re around 0 75.1%

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

      1. neg-mul-1 75.1%

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

      2. +-commutative 75.1%

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

      3. unpow2 75.1%

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

      4. times-frac 76.9%

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

      5. fma-udef 76.9%

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

      6. fma-neg 76.9%

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

      7. associate-*r/ 77.6%

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

      8. *-commutative 77.6%

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

      9. div-sub 79.9%

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

   7. Simplified 79.9%

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

   if 1.2000000000000001e-11 < y.re

   1. Initial program 46.5%

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

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

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. unsub-neg 80.9%

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

      4. unpow2 80.9%

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

      5. associate-/l* 82.7%

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

      6. associate-/r/ 82.7%

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

   4. Simplified 82.7%

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

   5. Taylor expanded in x.re around 0 80.9%

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

      1. unpow2 80.9%

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

      2. times-frac 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in x.im around 0 80.9%

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. *-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 89.1%

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

      6. distribute-lft-neg-in 89.1%

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

      7. cancel-sign-sub-inv 89.1%

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

      8. associate-*r/ 89.1%

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

      9. *-commutative 89.1%

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

      10. div-sub 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 77.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.5e-29)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 1.42e-11)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.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 <= -3.5e-29) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.42e-11) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= (-3.5d-29)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 1.42d-11) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_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 <= -3.5e-29) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.42e-11) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -3.5e-29:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 1.42e-11:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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 <= -3.5e-29)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.42e-11)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_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 <= -3.5e-29)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.42e-11)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_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, -3.5e-29], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.42e-11], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.4999999999999997e-29

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/l* 68.6%

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

      6. associate-/r/ 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in x.re around 0 74.3%

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

      1. unpow2 74.3%

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

      2. times-frac 82.4%

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

   7. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   9. Applied egg-rr 83.7%

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

   if -3.4999999999999997e-29 < y.re < 1.42e-11

   1. Initial program 75.2%

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

   2. Taylor expanded in y.re around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

      6. times-frac 76.9%

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

   4. Simplified 76.9%

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

      1. associate-*r/ 80.5%

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

      2. sub-div 82.8%

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

   6. Applied egg-rr 82.8%

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

   if 1.42e-11 < y.re

   1. Initial program 46.5%

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

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

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. unsub-neg 80.9%

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

      4. unpow2 80.9%

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

      5. associate-/l* 82.7%

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

      6. associate-/r/ 82.7%

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

   4. Simplified 82.7%

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

   5. Taylor expanded in x.re around 0 80.9%

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

      1. unpow2 80.9%

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

      2. times-frac 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in x.im around 0 80.9%

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. *-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 89.1%

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

      6. distribute-lft-neg-in 89.1%

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

      7. cancel-sign-sub-inv 89.1%

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

      8. associate-*r/ 89.1%

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

      9. *-commutative 89.1%

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

      10. div-sub 89.1%

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

   10. Simplified 89.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 10: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.22 \cdot 10^{+15} \lor \neg \left(y.im \leq 1.7 \cdot 10^{-104} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+42}\right) \land y.im \leq 3.7 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.22e+15)
         (not
          (or (<= y.im 1.7e-104)
              (and (not (<= y.im 7.2e+42)) (<= y.im 3.7e+84)))))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.22e+15) || !((y_46_im <= 1.7e-104) || (!(y_46_im <= 7.2e+42) && (y_46_im <= 3.7e+84)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.22d+15)) .or. (.not. (y_46im <= 1.7d-104) .or. (.not. (y_46im <= 7.2d+42)) .and. (y_46im <= 3.7d+84))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.22e+15) || !((y_46_im <= 1.7e-104) || (!(y_46_im <= 7.2e+42) && (y_46_im <= 3.7e+84)))) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.22e+15) or not ((y_46_im <= 1.7e-104) or (not (y_46_im <= 7.2e+42) and (y_46_im <= 3.7e+84))):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.22e+15) || !((y_46_im <= 1.7e-104) || (!(y_46_im <= 7.2e+42) && (y_46_im <= 3.7e+84))))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.22e+15) || ~(((y_46_im <= 1.7e-104) || (~((y_46_im <= 7.2e+42)) && (y_46_im <= 3.7e+84)))))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.22e+15], N[Not[Or[LessEqual[y$46$im, 1.7e-104], And[N[Not[LessEqual[y$46$im, 7.2e+42]], $MachinePrecision], LessEqual[y$46$im, 3.7e+84]]]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.22e15 or 1.70000000000000008e-104 < y.im < 7.2000000000000002e42 or 3.7e84 < y.im

   1. Initial program 55.1%

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

   2. Taylor expanded in y.re around 0 61.5%

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

      1. associate-*r/ 61.5%

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

      2. neg-mul-1 61.5%

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

   4. Simplified 61.5%

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

   if -1.22e15 < y.im < 1.70000000000000008e-104 or 7.2000000000000002e42 < y.im < 3.7e84

   1. Initial program 70.5%

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

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

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.22 \cdot 10^{+15} \lor \neg \left(y.im \leq 1.7 \cdot 10^{-104} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+42}\right) \land y.im \leq 3.7 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in y.re around inf 45.5%

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

3. Final simplification 45.5%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))