_divideComplex, imaginary part

Percentage Accurate: 61.5% → 89.0%

Time: 13.6s

Alternatives: 12

Speedup: 1.8×

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: 61.5% 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: 89.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}\\ t_2 := t_0 - y.im \cdot \frac{x.re}{t_1}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{\frac{t_1}{y.re}} - \frac{y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im))))
        (t_1 (pow (hypot y.re y.im) 2.0))
        (t_2 (- t_0 (* y.im (/ x.re t_1)))))
   (if (<= y.im -3.8e+101)
     (-
      (/ x.im (/ t_1 y.re))
      (/ (* y.im (/ x.re (hypot y.re y.im))) (hypot y.re y.im)))
     (if (<= y.im -2.9e-113)
       t_2
       (if (<= y.im 2.3e-163)
         (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
         (if (<= y.im 4.8e+87) t_2 (- t_0 (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	double t_1 = pow(hypot(y_46_re, y_46_im), 2.0);
	double t_2 = t_0 - (y_46_im * (x_46_re / t_1));
	double tmp;
	if (y_46_im <= -3.8e+101) {
		tmp = (x_46_im / (t_1 / y_46_re)) - ((y_46_im * (x_46_re / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -2.9e-113) {
		tmp = t_2;
	} else if (y_46_im <= 2.3e-163) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 4.8e+87) {
		tmp = t_2;
	} else {
		tmp = t_0 - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	double t_1 = Math.pow(Math.hypot(y_46_re, y_46_im), 2.0);
	double t_2 = t_0 - (y_46_im * (x_46_re / t_1));
	double tmp;
	if (y_46_im <= -3.8e+101) {
		tmp = (x_46_im / (t_1 / y_46_re)) - ((y_46_im * (x_46_re / Math.hypot(y_46_re, y_46_im))) / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -2.9e-113) {
		tmp = t_2;
	} else if (y_46_im <= 2.3e-163) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 4.8e+87) {
		tmp = t_2;
	} else {
		tmp = t_0 - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	t_1 = math.pow(math.hypot(y_46_re, y_46_im), 2.0)
	t_2 = t_0 - (y_46_im * (x_46_re / t_1))
	tmp = 0
	if y_46_im <= -3.8e+101:
		tmp = (x_46_im / (t_1 / y_46_re)) - ((y_46_im * (x_46_re / math.hypot(y_46_re, y_46_im))) / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -2.9e-113:
		tmp = t_2
	elif y_46_im <= 2.3e-163:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_im <= 4.8e+87:
		tmp = t_2
	else:
		tmp = t_0 - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)))
	t_1 = hypot(y_46_re, y_46_im) ^ 2.0
	t_2 = Float64(t_0 - Float64(y_46_im * Float64(x_46_re / t_1)))
	tmp = 0.0
	if (y_46_im <= -3.8e+101)
		tmp = Float64(Float64(x_46_im / Float64(t_1 / y_46_re)) - Float64(Float64(y_46_im * Float64(x_46_re / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -2.9e-113)
		tmp = t_2;
	elseif (y_46_im <= 2.3e-163)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 4.8e+87)
		tmp = t_2;
	else
		tmp = Float64(t_0 - 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)
	t_0 = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	t_1 = hypot(y_46_re, y_46_im) ^ 2.0;
	t_2 = t_0 - (y_46_im * (x_46_re / t_1));
	tmp = 0.0;
	if (y_46_im <= -3.8e+101)
		tmp = (x_46_im / (t_1 / y_46_re)) - ((y_46_im * (x_46_re / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -2.9e-113)
		tmp = t_2;
	elseif (y_46_im <= 2.3e-163)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_im <= 4.8e+87)
		tmp = t_2;
	else
		tmp = t_0 - (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_] := Block[{t$95$0 = N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(y$46$im * N[(x$46$re / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+101], N[(N[(x$46$im / N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.9e-113], t$95$2, If[LessEqual[y$46$im, 2.3e-163], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.8e+87], t$95$2, N[(t$95$0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.7999999999999998e101

   1. Initial program 39.5%

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

      1. div-sub 39.5%

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

      2. sub-neg 39.5%

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

      3. add-sqr-sqrt 39.5%

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

      4. pow2 39.5%

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

      5. hypot-def 39.5%

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

      6. associate-/l* 48.1%

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

      7. add-sqr-sqrt 48.1%

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

      8. pow2 48.1%

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

      9. hypot-def 48.1%

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

   4. Applied egg-rr 48.1%

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

      1. sub-neg 48.1%

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

      2. associate-/l* 46.5%

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

      3. associate-/r/ 44.6%

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

   6. Simplified 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. unpow2 44.6%

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

      3. times-frac 73.4%

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

   8. Applied egg-rr 73.4%

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

      1. associate-*l/ 73.6%

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

      2. *-lft-identity 73.6%

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

   10. Simplified 73.6%

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

       1. associate-*l/ 90.3%

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

   12. Applied egg-rr 90.3%

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

   if -3.7999999999999998e101 < y.im < -2.90000000000000004e-113 or 2.2999999999999999e-163 < y.im < 4.79999999999999963e87

   1. Initial program 72.1%

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

      1. div-sub 72.1%

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

      2. sub-neg 72.1%

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

      3. add-sqr-sqrt 72.1%

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

      4. pow2 72.1%

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

      5. hypot-def 72.1%

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

      6. associate-/l* 73.1%

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

      7. add-sqr-sqrt 73.1%

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

      8. pow2 73.1%

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

      9. hypot-def 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. sub-neg 73.1%

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

      2. associate-/l* 75.6%

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

      3. associate-/r/ 73.0%

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

   6. Simplified 73.0%

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

      1. div-inv 72.9%

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

      2. clear-num 72.9%

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

      3. div-inv 72.9%

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

      4. associate-*l* 70.3%

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

      5. *-commutative 70.3%

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

      6. div-inv 70.4%

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

      7. unpow2 70.4%

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

      8. times-frac 92.5%

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

   8. Applied egg-rr 92.5%

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

   if -2.90000000000000004e-113 < y.im < 2.2999999999999999e-163

   1. Initial program 65.1%

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

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

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   4. 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. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow2 74.2%

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

      2. div-inv 74.2%

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

      3. associate-*l* 87.1%

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

   7. Applied egg-rr 87.1%

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

   if 4.79999999999999963e87 < y.im

   1. Initial program 36.5%

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

      1. div-sub 36.5%

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

      2. sub-neg 36.5%

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

      3. add-sqr-sqrt 36.5%

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

      4. pow2 36.5%

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

      5. hypot-def 36.5%

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

      6. associate-/l* 39.8%

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

      7. add-sqr-sqrt 39.8%

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

      8. pow2 39.8%

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

      9. hypot-def 39.8%

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

   4. Applied egg-rr 39.8%

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

      1. sub-neg 39.8%

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

      2. associate-/l* 40.4%

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

      3. associate-/r/ 36.6%

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

   6. Simplified 36.6%

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

      1. div-inv 36.6%

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

      2. clear-num 36.5%

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

      3. div-inv 36.5%

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

      4. associate-*l* 36.0%

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

      5. *-commutative 36.0%

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

      6. div-inv 36.0%

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

      7. unpow2 36.0%

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

      8. times-frac 53.6%

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

   8. Applied egg-rr 53.6%

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

   9. Taylor expanded in y.re around 0 94.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - \frac{y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-113}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ t_2 := t_0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im))))
        (t_1 (- t_0 (* y.im (/ x.re (pow (hypot y.re y.im) 2.0)))))
        (t_2 (- t_0 (/ x.re y.im))))
   (if (<= y.im -6.8e+105)
     t_2
     (if (<= y.im -2.9e-113)
       t_1
       (if (<= y.im 6.6e-163)
         (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
         (if (<= y.im 4.8e+87) t_1 t_2))))))

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 / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	double t_1 = t_0 - (y_46_im * (x_46_re / pow(hypot(y_46_re, y_46_im), 2.0)));
	double t_2 = t_0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -6.8e+105) {
		tmp = t_2;
	} else if (y_46_im <= -2.9e-113) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e-163) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 4.8e+87) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	double t_1 = t_0 - (y_46_im * (x_46_re / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0)));
	double t_2 = t_0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -6.8e+105) {
		tmp = t_2;
	} else if (y_46_im <= -2.9e-113) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e-163) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 4.8e+87) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	t_1 = t_0 - (y_46_im * (x_46_re / math.pow(math.hypot(y_46_re, y_46_im), 2.0)))
	t_2 = t_0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -6.8e+105:
		tmp = t_2
	elif y_46_im <= -2.9e-113:
		tmp = t_1
	elif y_46_im <= 6.6e-163:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_im <= 4.8e+87:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)))
	t_1 = Float64(t_0 - Float64(y_46_im * Float64(x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0))))
	t_2 = Float64(t_0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -6.8e+105)
		tmp = t_2;
	elseif (y_46_im <= -2.9e-113)
		tmp = t_1;
	elseif (y_46_im <= 6.6e-163)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 4.8e+87)
		tmp = t_1;
	else
		tmp = t_2;
	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 / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	t_1 = t_0 - (y_46_im * (x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0)));
	t_2 = t_0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -6.8e+105)
		tmp = t_2;
	elseif (y_46_im <= -2.9e-113)
		tmp = t_1;
	elseif (y_46_im <= 6.6e-163)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_im <= 4.8e+87)
		tmp = t_1;
	else
		tmp = t_2;
	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 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(y$46$im * N[(x$46$re / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.8e+105], t$95$2, If[LessEqual[y$46$im, -2.9e-113], t$95$1, If[LessEqual[y$46$im, 6.6e-163], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.8e+87], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.7999999999999999e105 or 4.79999999999999963e87 < y.im

   1. Initial program 38.3%

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

      1. div-sub 38.3%

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

      2. sub-neg 38.3%

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

      3. add-sqr-sqrt 38.3%

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

      4. pow2 38.3%

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

      5. hypot-def 38.3%

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

      6. associate-/l* 44.4%

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

      7. add-sqr-sqrt 44.4%

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

      8. pow2 44.4%

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

      9. hypot-def 44.4%

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

   4. Applied egg-rr 44.4%

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

      1. sub-neg 44.4%

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

      2. associate-/l* 43.8%

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

      3. associate-/r/ 41.0%

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

   6. Simplified 41.0%

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

      1. div-inv 40.9%

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

      2. clear-num 40.9%

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

      3. div-inv 40.9%

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

      4. associate-*l* 41.5%

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

      5. *-commutative 41.5%

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

      6. div-inv 41.5%

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

      7. unpow2 41.5%

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

      8. times-frac 54.4%

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

   8. Applied egg-rr 54.4%

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

   9. Taylor expanded in y.re around 0 88.5%

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

   if -6.7999999999999999e105 < y.im < -2.90000000000000004e-113 or 6.60000000000000002e-163 < y.im < 4.79999999999999963e87

   1. Initial program 71.5%

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

      1. div-sub 71.5%

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

      2. sub-neg 71.5%

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

      3. add-sqr-sqrt 71.4%

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

      4. pow2 71.4%

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

      5. hypot-def 71.4%

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

      6. associate-/l* 72.4%

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

      7. add-sqr-sqrt 72.4%

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

      8. pow2 72.4%

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

      9. hypot-def 72.4%

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

   4. Applied egg-rr 72.4%

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

      1. sub-neg 72.4%

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

      2. associate-/l* 75.0%

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

      3. associate-/r/ 72.4%

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

   6. Simplified 72.4%

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

      1. div-inv 72.2%

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

      2. clear-num 72.2%

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

      3. div-inv 72.3%

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

      4. associate-*l* 69.7%

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

      5. *-commutative 69.7%

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

      6. div-inv 69.8%

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

      7. unpow2 69.8%

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

      8. times-frac 91.7%

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

   8. Applied egg-rr 91.7%

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

   if -2.90000000000000004e-113 < y.im < 6.60000000000000002e-163

   1. Initial program 65.1%

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

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

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   4. 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. associate-/l* 74.2%

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

   5. Simplified 74.2%

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

      1. pow2 74.2%

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

      2. div-inv 74.2%

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

      3. associate-*l* 87.1%

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

   7. Applied egg-rr 87.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-113}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ t_1 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{\frac{t_1}{y.re}} - y.im \cdot \frac{x.re}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
          (/ x.re y.im)))
        (t_1 (pow (hypot y.re y.im) 2.0)))
   (if (<= y.im -4.8e+91)
     t_0
     (if (<= y.im -9.8e-104)
       (-
        (/ x.im y.re)
        (* y.im (/ (/ x.re (hypot y.re y.im)) (hypot y.re y.im))))
       (if (<= y.im 9e-70)
         (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
         (if (<= y.im 1.6e+87)
           (- (/ x.im (/ t_1 y.re)) (* y.im (/ x.re t_1)))
           t_0))))))

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 / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	double t_1 = pow(hypot(y_46_re, y_46_im), 2.0);
	double tmp;
	if (y_46_im <= -4.8e+91) {
		tmp = t_0;
	} else if (y_46_im <= -9.8e-104) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	} else if (y_46_im <= 9e-70) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.6e+87) {
		tmp = (x_46_im / (t_1 / y_46_re)) - (y_46_im * (x_46_re / t_1));
	} else {
		tmp = t_0;
	}
	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 / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	double t_1 = Math.pow(Math.hypot(y_46_re, y_46_im), 2.0);
	double tmp;
	if (y_46_im <= -4.8e+91) {
		tmp = t_0;
	} else if (y_46_im <= -9.8e-104) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im)));
	} else if (y_46_im <= 9e-70) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.6e+87) {
		tmp = (x_46_im / (t_1 / y_46_re)) - (y_46_im * (x_46_re / t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im)
	t_1 = math.pow(math.hypot(y_46_re, y_46_im), 2.0)
	tmp = 0
	if y_46_im <= -4.8e+91:
		tmp = t_0
	elif y_46_im <= -9.8e-104:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)))
	elif y_46_im <= 9e-70:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_im <= 1.6e+87:
		tmp = (x_46_im / (t_1 / y_46_re)) - (y_46_im * (x_46_re / t_1))
	else:
		tmp = t_0
	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 / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(x_46_re / y_46_im))
	t_1 = hypot(y_46_re, y_46_im) ^ 2.0
	tmp = 0.0
	if (y_46_im <= -4.8e+91)
		tmp = t_0;
	elseif (y_46_im <= -9.8e-104)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))));
	elseif (y_46_im <= 9e-70)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 1.6e+87)
		tmp = Float64(Float64(x_46_im / Float64(t_1 / y_46_re)) - Float64(y_46_im * Float64(x_46_re / t_1)));
	else
		tmp = t_0;
	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 / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	t_1 = hypot(y_46_re, y_46_im) ^ 2.0;
	tmp = 0.0;
	if (y_46_im <= -4.8e+91)
		tmp = t_0;
	elseif (y_46_im <= -9.8e-104)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 9e-70)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_im <= 1.6e+87)
		tmp = (x_46_im / (t_1 / y_46_re)) - (y_46_im * (x_46_re / t_1));
	else
		tmp = t_0;
	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 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+91], t$95$0, If[LessEqual[y$46$im, -9.8e-104], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9e-70], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+87], N[(N[(x$46$im / N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(x$46$re / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.79999999999999966e91 or 1.6e87 < y.im

   1. Initial program 39.8%

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

      1. div-sub 39.8%

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

      2. sub-neg 39.8%

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

      3. add-sqr-sqrt 39.8%

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

      4. pow2 39.8%

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

      5. hypot-def 39.8%

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

      6. associate-/l* 45.6%

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

      7. add-sqr-sqrt 45.6%

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

      8. pow2 45.6%

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

      9. hypot-def 45.6%

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

   4. Applied egg-rr 45.6%

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

      1. sub-neg 45.6%

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

      2. associate-/l* 45.1%

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

      3. associate-/r/ 42.3%

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

   6. Simplified 42.3%

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

      1. div-inv 42.3%

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

      2. clear-num 42.3%

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

      3. div-inv 42.3%

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

      4. associate-*l* 42.8%

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

      5. *-commutative 42.8%

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

      6. div-inv 42.8%

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

      7. unpow2 42.8%

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

      8. times-frac 55.2%

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

   8. Applied egg-rr 55.2%

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

   9. Taylor expanded in y.re around 0 88.0%

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

   if -4.79999999999999966e91 < y.im < -9.8000000000000006e-104

   1. Initial program 73.3%

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

      1. div-sub 73.3%

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

      2. sub-neg 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. pow2 73.3%

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

      5. hypot-def 73.3%

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

      6. associate-/l* 76.2%

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

      7. add-sqr-sqrt 76.2%

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

      8. pow2 76.2%

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

      9. hypot-def 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. sub-neg 76.2%

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

      2. associate-/l* 77.0%

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

      3. associate-/r/ 74.3%

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

   6. Simplified 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 74.3%

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

   8. Applied egg-rr 74.3%

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

      1. associate-*l/ 74.4%

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

      2. *-lft-identity 74.4%

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

   10. Simplified 74.4%

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

   11. Taylor expanded in y.re around inf 83.7%

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

   if -9.8000000000000006e-104 < y.im < 9.00000000000000044e-70

   1. Initial program 64.0%

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

   3. Taylor expanded in y.re around inf 75.8%

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

      1. +-commutative 75.8%

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

      2. mul-1-neg 75.8%

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

      3. unsub-neg 75.8%

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

      4. associate-/l* 76.2%

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

   5. Simplified 76.2%

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

      1. pow2 76.2%

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

      2. div-inv 76.2%

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

      3. associate-*l* 84.8%

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

   7. Applied egg-rr 84.8%

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

   if 9.00000000000000044e-70 < y.im < 1.6e87

   1. Initial program 75.8%

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

      1. div-sub 75.8%

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

      2. sub-neg 75.8%

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

      3. add-sqr-sqrt 75.8%

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

      4. pow2 75.8%

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

      5. hypot-def 75.8%

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

      6. associate-/l* 75.9%

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

      7. add-sqr-sqrt 75.9%

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

      8. pow2 75.9%

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

      9. hypot-def 75.9%

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

   4. Applied egg-rr 75.9%

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

      1. sub-neg 75.9%

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

      2. associate-/l* 78.9%

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

      3. associate-/r/ 78.8%

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

   6. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (/ x.im y.re)
          (* y.im (/ (/ x.re (hypot y.re y.im)) (hypot y.re y.im)))))
        (t_1
         (-
          (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
          (/ x.re y.im))))
   (if (<= y.im -4.8e+91)
     t_1
     (if (<= y.im -1.9e-103)
       t_0
       (if (<= y.im 4.5e-95)
         (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
         (if (<= y.im 1.45e+48) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	double t_1 = ((y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.8e+91) {
		tmp = t_1;
	} else if (y_46_im <= -1.9e-103) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-95) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.45e+48) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im)));
	double t_1 = ((y_46_re / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.8e+91) {
		tmp = t_1;
	} else if (y_46_im <= -1.9e-103) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-95) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.45e+48) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)))
	t_1 = ((y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -4.8e+91:
		tmp = t_1
	elif y_46_im <= -1.9e-103:
		tmp = t_0
	elif y_46_im <= 4.5e-95:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_im <= 1.45e+48:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))))
	t_1 = Float64(Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e+91)
		tmp = t_1;
	elseif (y_46_im <= -1.9e-103)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-95)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 1.45e+48)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)));
	t_1 = ((y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4.8e+91)
		tmp = t_1;
	elseif (y_46_im <= -1.9e-103)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-95)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_im <= 1.45e+48)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+91], t$95$1, If[LessEqual[y$46$im, -1.9e-103], t$95$0, If[LessEqual[y$46$im, 4.5e-95], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.45e+48], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.79999999999999966e91 or 1.4499999999999999e48 < y.im

   1. Initial program 41.6%

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

      1. div-sub 41.6%

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

      2. sub-neg 41.6%

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

      3. add-sqr-sqrt 41.6%

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

      4. pow2 41.6%

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

      5. hypot-def 41.6%

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

      6. associate-/l* 47.2%

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

      7. add-sqr-sqrt 47.2%

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

      8. pow2 47.2%

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

      9. hypot-def 47.2%

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

   4. Applied egg-rr 47.2%

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

      1. sub-neg 47.2%

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

      2. associate-/l* 46.7%

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

      3. associate-/r/ 44.1%

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

   6. Simplified 44.1%

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

      1. div-inv 44.1%

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

      2. clear-num 44.1%

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

      3. div-inv 44.1%

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

      4. associate-*l* 44.6%

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

      5. *-commutative 44.6%

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

      6. div-inv 44.5%

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

      7. unpow2 44.5%

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

      8. times-frac 57.3%

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

   8. Applied egg-rr 57.3%

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

   9. Taylor expanded in y.re around 0 87.7%

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

   if -4.79999999999999966e91 < y.im < -1.9e-103 or 4.5e-95 < y.im < 1.4499999999999999e48

   1. Initial program 71.9%

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

      1. div-sub 71.9%

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

      2. sub-neg 71.9%

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

      3. add-sqr-sqrt 71.9%

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

      4. pow2 71.9%

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

      5. hypot-def 71.9%

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

      6. associate-/l* 73.2%

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

      7. add-sqr-sqrt 73.2%

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

      8. pow2 73.2%

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

      9. hypot-def 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. sub-neg 73.2%

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

      2. associate-/l* 76.5%

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

      3. associate-/r/ 75.2%

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

   6. Simplified 75.2%

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

      1. *-un-lft-identity 75.2%

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

      2. unpow2 75.2%

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

      3. times-frac 77.8%

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

   8. Applied egg-rr 77.8%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

   10. Simplified 78.0%

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

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

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

   if -1.9e-103 < y.im < 4.5e-95

   1. Initial program 65.7%

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

   3. Taylor expanded in y.re around inf 75.4%

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

      1. +-commutative 75.4%

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

      2. mul-1-neg 75.4%

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

      3. unsub-neg 75.4%

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

      4. associate-/l* 75.8%

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

   5. Simplified 75.8%

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

      1. pow2 75.8%

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

      2. div-inv 75.8%

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

      3. associate-*l* 85.0%

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

   7. Applied egg-rr 85.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
          (/ x.re y.im))))
   (if (<= y.im -1.65e-31)
     t_0
     (if (<= y.im 5.6e-72)
       (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
       (if (<= y.im 3.8e+14)
         (/ (fma x.im y.re (* y.im (- x.re))) (fma y.re y.re (* y.im y.im)))
         t_0)))))

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 / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.65e-31) {
		tmp = t_0;
	} else if (y_46_im <= 5.6e-72) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 3.8e+14) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	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 / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.65e-31)
		tmp = t_0;
	elseif (y_46_im <= 5.6e-72)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 3.8e+14)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e-31], t$95$0, If[LessEqual[y$46$im, 5.6e-72], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+14], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65e-31 or 3.8e14 < y.im

   1. Initial program 45.1%

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

      1. div-sub 45.1%

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

      2. sub-neg 45.1%

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

      3. add-sqr-sqrt 45.1%

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

      4. pow2 45.1%

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

      5. hypot-def 45.1%

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

      6. associate-/l* 50.4%

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

      7. add-sqr-sqrt 50.4%

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

      8. pow2 50.4%

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

      9. hypot-def 50.4%

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

   4. Applied egg-rr 50.4%

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

      1. sub-neg 50.4%

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

      2. associate-/l* 50.2%

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

      3. associate-/r/ 47.4%

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

   6. Simplified 47.4%

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

      1. div-inv 47.4%

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

      2. clear-num 47.4%

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

      3. div-inv 47.4%

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

      4. associate-*l* 47.5%

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

      5. *-commutative 47.5%

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

      6. div-inv 47.5%

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

      7. unpow2 47.5%

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

      8. times-frac 63.8%

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

   8. Applied egg-rr 63.8%

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

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

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

   if -1.65e-31 < y.im < 5.5999999999999996e-72

   1. Initial program 66.4%

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

   3. Taylor expanded in y.re around inf 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. associate-/l* 74.7%

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

   5. Simplified 74.7%

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

      1. pow2 74.7%

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

      2. div-inv 74.7%

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

      3. associate-*l* 82.0%

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

   7. Applied egg-rr 82.0%

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

   if 5.5999999999999996e-72 < y.im < 3.8e14

   1. Initial program 83.5%

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

      1. fma-neg 83.5%

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

      2. distribute-lft-neg-out 83.5%

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

      3. *-commutative 83.5%

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

      4. fma-def 83.5%

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

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-31}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\frac{{y.im}^{2}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{-71}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.8e+91)
   (- (/ y.re (/ (pow y.im 2.0) x.im)) (/ x.re y.im))
   (if (<= y.im 1e-71)
     (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
     (if (<= y.im 2e+45)
       (/ (fma x.im y.re (* y.im (- x.re))) (fma y.re y.re (* y.im y.im)))
       (- (/ x.im (* y.im (/ y.im 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_im <= -4.8e+91) {
		tmp = (y_46_re / (pow(y_46_im, 2.0) / x_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 1e-71) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 2e+45) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.8e+91)
		tmp = Float64(Float64(y_46_re / Float64((y_46_im ^ 2.0) / x_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 1e-71)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 2e+45)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.8e+91], N[(N[(y$46$re / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1e-71], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+45], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.79999999999999966e91

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. mul-1-neg 74.7%

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

      3. unsub-neg 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

   if -4.79999999999999966e91 < y.im < 9.9999999999999992e-72

   1. Initial program 66.7%

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

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

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

      1. +-commutative 72.7%

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

      2. mul-1-neg 72.7%

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

      3. unsub-neg 72.7%

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

      4. associate-/l* 72.1%

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

   5. Simplified 72.1%

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

      1. pow2 72.1%

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

      2. div-inv 72.2%

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

      3. associate-*l* 78.2%

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

   7. Applied egg-rr 78.2%

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

   if 9.9999999999999992e-72 < y.im < 1.9999999999999999e45

   1. Initial program 76.7%

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

      1. fma-neg 76.8%

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

      2. distribute-lft-neg-out 76.8%

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

      3. *-commutative 76.8%

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

      4. fma-def 76.8%

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

   3. Simplified 76.8%

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

   if 1.9999999999999999e45 < y.im

   1. Initial program 40.8%

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

      1. div-sub 40.8%

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

      2. sub-neg 40.8%

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

      3. add-sqr-sqrt 40.8%

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

      4. pow2 40.8%

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

      5. hypot-def 40.8%

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

      6. associate-/l* 43.7%

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

      7. add-sqr-sqrt 43.7%

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

      8. pow2 43.7%

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

      9. hypot-def 43.7%

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

   4. Applied egg-rr 43.7%

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

      1. sub-neg 43.7%

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

      2. associate-/l* 44.3%

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

      3. associate-/r/ 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in y.re around 0 76.6%

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

   8. Taylor expanded in y.re around 0 76.6%

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

      1. pow2 76.6%

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

      2. *-un-lft-identity 76.6%

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

      3. times-frac 78.4%

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

   10. Applied egg-rr 78.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\frac{{y.im}^{2}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10^{-71}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\frac{{y.im}^{2}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.02 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.8e+91)
   (- (/ y.re (/ (pow y.im 2.0) x.im)) (/ x.re y.im))
   (if (<= y.im 2.02e-72)
     (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))
     (if (<= y.im 1.16e+45)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (- (/ x.im (* y.im (/ y.im 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_im <= -4.8e+91) {
		tmp = (y_46_re / (pow(y_46_im, 2.0) / x_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 2.02e-72) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.16e+45) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-4.8d+91)) then
        tmp = (y_46re / ((y_46im ** 2.0d0) / x_46im)) - (x_46re / y_46im)
    else if (y_46im <= 2.02d-72) then
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re * (1.0d0 / y_46im))))
    else if (y_46im <= 1.16d+45) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else
        tmp = (x_46im / (y_46im * (y_46im / y_46re))) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.8e+91) {
		tmp = (y_46_re / (Math.pow(y_46_im, 2.0) / x_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_im <= 2.02e-72) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_im <= 1.16e+45) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.8e+91:
		tmp = (y_46_re / (math.pow(y_46_im, 2.0) / x_46_im)) - (x_46_re / y_46_im)
	elif y_46_im <= 2.02e-72:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_im <= 1.16e+45:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	else:
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.8e+91)
		tmp = Float64(Float64(y_46_re / Float64((y_46_im ^ 2.0) / x_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 2.02e-72)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_im <= 1.16e+45)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.8e+91)
		tmp = (y_46_re / ((y_46_im ^ 2.0) / x_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_im <= 2.02e-72)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_im <= 1.16e+45)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	else
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.8e+91], N[(N[(y$46$re / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.02e-72], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.16e+45], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.79999999999999966e91

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0 74.7%

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

      1. +-commutative 74.7%

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

      2. mul-1-neg 74.7%

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

      3. unsub-neg 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

   if -4.79999999999999966e91 < y.im < 2.02e-72

   1. Initial program 66.7%

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

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

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

      1. +-commutative 72.7%

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

      2. mul-1-neg 72.7%

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

      3. unsub-neg 72.7%

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

      4. associate-/l* 72.1%

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

   5. Simplified 72.1%

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

      1. pow2 72.1%

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

      2. div-inv 72.2%

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

      3. associate-*l* 78.2%

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

   7. Applied egg-rr 78.2%

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

   if 2.02e-72 < y.im < 1.1600000000000001e45

   1. Initial program 76.7%

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

   if 1.1600000000000001e45 < y.im

   1. Initial program 40.8%

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

      1. div-sub 40.8%

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

      2. sub-neg 40.8%

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

      3. add-sqr-sqrt 40.8%

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

      4. pow2 40.8%

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

      5. hypot-def 40.8%

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

      6. associate-/l* 43.7%

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

      7. add-sqr-sqrt 43.7%

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

      8. pow2 43.7%

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

      9. hypot-def 43.7%

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

   4. Applied egg-rr 43.7%

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

      1. sub-neg 43.7%

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

      2. associate-/l* 44.3%

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

      3. associate-/r/ 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in y.re around 0 76.6%

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

   8. Taylor expanded in y.re around 0 76.6%

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

      1. pow2 76.6%

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

      2. *-un-lft-identity 76.6%

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

      3. times-frac 78.4%

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

   10. Applied egg-rr 78.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re}{\frac{{y.im}^{2}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.02 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))))
   (if (<= y.re -9e+88)
     t_0
     (if (<= y.re 4.7e-157)
       (- (/ x.im (* y.im (/ y.im y.re))) (/ x.re y.im))
       (if (<= y.re 5.2e+126)
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	double tmp;
	if (y_46_re <= -9e+88) {
		tmp = t_0;
	} else if (y_46_re <= 4.7e-157) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 5.2e+126) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re * (1.0d0 / y_46im))))
    if (y_46re <= (-9d+88)) then
        tmp = t_0
    else if (y_46re <= 4.7d-157) then
        tmp = (x_46im / (y_46im * (y_46im / y_46re))) - (x_46re / y_46im)
    else if (y_46re <= 5.2d+126) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	double tmp;
	if (y_46_re <= -9e+88) {
		tmp = t_0;
	} else if (y_46_re <= 4.7e-157) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else if (y_46_re <= 5.2e+126) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	tmp = 0
	if y_46_re <= -9e+88:
		tmp = t_0
	elif y_46_re <= 4.7e-157:
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	elif y_46_re <= 5.2e+126:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))))
	tmp = 0.0
	if (y_46_re <= -9e+88)
		tmp = t_0;
	elseif (y_46_re <= 4.7e-157)
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 5.2e+126)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	tmp = 0.0;
	if (y_46_re <= -9e+88)
		tmp = t_0;
	elseif (y_46_re <= 4.7e-157)
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	elseif (y_46_re <= 5.2e+126)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+88], t$95$0, If[LessEqual[y$46$re, 4.7e-157], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+126], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9e88 or 5.1999999999999999e126 < y.re

   1. Initial program 33.8%

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

   3. Taylor expanded in y.re around inf 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

      4. associate-/l* 72.3%

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

   5. Simplified 72.3%

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

      1. pow2 72.3%

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

      2. div-inv 72.3%

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

      3. associate-*l* 78.4%

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

   7. Applied egg-rr 78.4%

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

   if -9e88 < y.re < 4.7000000000000002e-157

   1. Initial program 66.0%

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

      1. div-sub 61.2%

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

      2. sub-neg 61.2%

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

      3. add-sqr-sqrt 61.1%

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

      4. pow2 61.1%

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

      5. hypot-def 61.1%

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

      6. associate-/l* 67.0%

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

      7. add-sqr-sqrt 67.0%

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

      8. pow2 67.0%

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

      9. hypot-def 67.0%

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

   4. Applied egg-rr 67.0%

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

      1. sub-neg 67.0%

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

      2. associate-/l* 67.2%

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

      3. associate-/r/ 62.9%

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

   6. Simplified 62.9%

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

   7. Taylor expanded in y.re around 0 78.2%

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

   8. Taylor expanded in y.re around 0 73.9%

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

      1. pow2 73.9%

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

      2. *-un-lft-identity 73.9%

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

      3. times-frac 78.4%

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

   10. Applied egg-rr 78.4%

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

   if 4.7000000000000002e-157 < y.re < 5.1999999999999999e126

   1. Initial program 78.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+92} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.9e+92) (not (<= y.im 1.2e+45)))
   (/ (- x.re) y.im)
   (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.9e+92) || !(y_46_im <= 1.2e+45)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.9d+92)) .or. (.not. (y_46im <= 1.2d+45))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re * (1.0d0 / y_46im))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.9e+92) || !(y_46_im <= 1.2e+45)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.9e+92) or not (y_46_im <= 1.2e+45):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.9e+92) || !(y_46_im <= 1.2e+45))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.9e+92) || ~((y_46_im <= 1.2e+45)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / 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$46$im, -4.9e+92], N[Not[LessEqual[y$46$im, 1.2e+45]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.9000000000000002e92 or 1.19999999999999995e45 < y.im

   1. Initial program 41.3%

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

   3. Taylor expanded in y.re around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if -4.9000000000000002e92 < y.im < 1.19999999999999995e45

   1. Initial program 69.3%

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

   3. Taylor expanded in y.re around inf 69.5%

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

      1. +-commutative 69.5%

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

      2. mul-1-neg 69.5%

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

      3. unsub-neg 69.5%

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

      4. associate-/l* 69.0%

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

   5. Simplified 69.0%

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

      1. pow2 69.0%

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

      2. div-inv 69.1%

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

      3. associate-*l* 73.7%

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

   7. Applied egg-rr 73.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+92} \lor \neg \left(y.im \leq 1.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+91} \lor \neg \left(y.im \leq 2.7 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7e+91) (not (<= y.im 2.7e+42)))
   (- (/ x.im (* y.im (/ y.im y.re))) (/ x.re y.im))
   (- (/ x.im y.re) (/ x.re (* y.re (* y.re (/ 1.0 y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e+91) || !(y_46_im <= 2.7e+42)) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7d+91)) .or. (.not. (y_46im <= 2.7d+42))) then
        tmp = (x_46im / (y_46im * (y_46im / y_46re))) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re * (1.0d0 / y_46im))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7e+91) || !(y_46_im <= 2.7e+42)) {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7e+91) or not (y_46_im <= 2.7e+42):
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / y_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7e+91) || !(y_46_im <= 2.7e+42))
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7e+91) || ~((y_46_im <= 2.7e+42)))
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re * (1.0 / 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$46$im, -7e+91], N[Not[LessEqual[y$46$im, 2.7e+42]], $MachinePrecision]], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.00000000000000001e91 or 2.7000000000000001e42 < y.im

   1. Initial program 41.8%

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

      1. div-sub 41.8%

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

      2. sub-neg 41.8%

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

      3. add-sqr-sqrt 41.8%

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

      4. pow2 41.8%

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

      5. hypot-def 41.8%

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

      6. associate-/l* 47.3%

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

      7. add-sqr-sqrt 47.3%

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

      8. pow2 47.3%

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

      9. hypot-def 47.3%

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

   4. Applied egg-rr 47.3%

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

      1. sub-neg 47.3%

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

      2. associate-/l* 46.8%

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

      3. associate-/r/ 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in y.re around 0 75.8%

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

   8. Taylor expanded in y.re around 0 75.8%

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

      1. pow2 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 76.8%

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

   10. Applied egg-rr 76.8%

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

   if -7.00000000000000001e91 < y.im < 2.7000000000000001e42

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

      4. associate-/l* 69.5%

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

   5. Simplified 69.5%

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

      1. pow2 69.5%

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

      2. div-inv 69.5%

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

      3. associate-*l* 74.1%

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

   7. Applied egg-rr 74.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+91} \lor \neg \left(y.im \leq 2.7 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+44} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.65e+44) (not (<= y.im 4.6e+43)))
   (/ (- 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 <= -1.65e+44) || !(y_46_im <= 4.6e+43)) {
		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 <= (-1.65d+44)) .or. (.not. (y_46im <= 4.6d+43))) 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 <= -1.65e+44) || !(y_46_im <= 4.6e+43)) {
		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 <= -1.65e+44) or not (y_46_im <= 4.6e+43):
		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 <= -1.65e+44) || !(y_46_im <= 4.6e+43))
		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 <= -1.65e+44) || ~((y_46_im <= 4.6e+43)))
		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, -1.65e+44], N[Not[LessEqual[y$46$im, 4.6e+43]], $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 < -1.65000000000000007e44 or 4.6000000000000005e43 < y.im

   1. Initial program 44.1%

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

   3. Taylor expanded in y.re around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. neg-mul-1 68.6%

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

   5. Simplified 68.6%

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

   if -1.65000000000000007e44 < y.im < 4.6000000000000005e43

   1. Initial program 68.7%

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

   3. Taylor expanded in y.re around inf 62.9%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+44} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.1% 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 57.3%

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

3. Taylor expanded in y.re around inf 41.2%

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

4. Final simplification 41.2%

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

Reproduce

herbie shell --seed 2024010 
(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))))