_divideComplex, imaginary part

Percentage Accurate: 62.1% → 89.0%

Time: 12.0s

Alternatives: 11

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: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 1.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1.4e+237)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (-
      (/ (* x.im (/ y.re (hypot y.im y.re))) (hypot 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 t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1.4e+237) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im * (y_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (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 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1.4e+237) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im * (y_46_re / Math.hypot(y_46_im, y_46_re))) / Math.hypot(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):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1.4e+237:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((x_46_im * (y_46_re / math.hypot(y_46_im, y_46_re))) / math.hypot(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)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1.4e+237)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / hypot(y_46_im, y_46_re))) / hypot(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)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1.4e+237)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((x_46_im * (y_46_re / hypot(y_46_im, y_46_re))) / hypot(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_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.4e+237], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.39999999999999992e237

   1. Initial program 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 77.3%

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

      4. hypot-def 77.3%

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

      5. hypot-def 96.2%

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

   3. Applied egg-rr 96.2%

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

   if 1.39999999999999992e237 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 19.3%

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

      1. div-sub 14.1%

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

      2. *-un-lft-identity 14.1%

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

      3. add-sqr-sqrt 14.1%

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

      4. times-frac 14.1%

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

      5. fma-neg 14.1%

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

      6. hypot-def 14.1%

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

      7. hypot-def 17.7%

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

      8. associate-/l* 24.0%

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

      9. add-sqr-sqrt 24.0%

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

      10. pow2 24.0%

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

      11. hypot-def 24.0%

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

   3. Applied egg-rr 24.0%

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

      1. fma-neg 24.0%

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

      2. *-commutative 24.0%

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

      3. associate-/l* 57.0%

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

      4. associate-/l* 49.4%

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

      5. *-commutative 49.4%

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

      6. associate-/l* 57.0%

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

   5. Simplified 57.0%

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

      1. associate-*l/ 57.1%

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

      2. *-un-lft-identity 57.1%

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

      3. associate-/r/ 57.1%

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

      4. hypot-udef 24.0%

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

      5. +-commutative 24.0%

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

      6. hypot-def 57.1%

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

      7. hypot-udef 24.0%

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

      8. +-commutative 24.0%

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

      9. hypot-def 57.1%

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

   7. Applied egg-rr 57.1%

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

   8. Taylor expanded in y.im around inf 80.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 1.4 \cdot 10^{+237}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 85.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1
         (-
          (/ (* x.im (/ y.re (hypot y.im y.re))) (hypot y.im y.re))
          (/ x.re y.im))))
   (if (<= y.im -5e+65)
     t_1
     (if (<= y.im -9.8e-79)
       t_0
       (if (<= y.im 3.2e-127)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 3e+15) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im * (y_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -5e+65) {
		tmp = t_1;
	} else if (y_46_im <= -9.8e-79) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-127) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 3e+15) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im * (y_46_re / Math.hypot(y_46_im, y_46_re))) / Math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -5e+65) {
		tmp = t_1;
	} else if (y_46_im <= -9.8e-79) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-127) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 3e+15) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((x_46_im * (y_46_re / math.hypot(y_46_im, y_46_re))) / math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -5e+65:
		tmp = t_1
	elif y_46_im <= -9.8e-79:
		tmp = t_0
	elif y_46_im <= 3.2e-127:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_im <= 3e+15:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_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)))
	t_1 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -5e+65)
		tmp = t_1;
	elseif (y_46_im <= -9.8e-79)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-127)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 3e+15)
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((x_46_im * (y_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5e+65)
		tmp = t_1;
	elseif (y_46_im <= -9.8e-79)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-127)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_im <= 3e+15)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$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]}, Block[{t$95$1 = N[(N[(N[(x$46$im * N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5e+65], t$95$1, If[LessEqual[y$46$im, -9.8e-79], t$95$0, If[LessEqual[y$46$im, 3.2e-127], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3e+15], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.99999999999999973e65 or 3e15 < y.im

   1. Initial program 32.6%

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

      1. div-sub 32.6%

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

      2. *-un-lft-identity 32.6%

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

      3. add-sqr-sqrt 32.6%

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

      4. times-frac 32.6%

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

      5. fma-neg 32.6%

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

      6. hypot-def 32.6%

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

      7. hypot-def 38.2%

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

      8. associate-/l* 44.5%

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

      9. add-sqr-sqrt 44.5%

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

      10. pow2 44.5%

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

      11. hypot-def 44.5%

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

   3. Applied egg-rr 44.5%

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

      1. fma-neg 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-/l* 60.4%

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

      4. associate-/l* 53.2%

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

      5. *-commutative 53.2%

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

      6. associate-/l* 59.5%

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

   5. Simplified 59.5%

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

      1. associate-*l/ 59.6%

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

      2. *-un-lft-identity 59.6%

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

      3. associate-/r/ 59.6%

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

      4. hypot-udef 39.4%

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

      5. +-commutative 39.4%

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

      6. hypot-def 59.6%

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

      7. hypot-udef 39.4%

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

      8. +-commutative 39.4%

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

      9. hypot-def 59.6%

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

   7. Applied egg-rr 59.6%

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

   8. Taylor expanded in y.im around inf 91.0%

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

   if -4.99999999999999973e65 < y.im < -9.8000000000000001e-79 or 3.20000000000000017e-127 < y.im < 3e15

   1. Initial program 87.7%

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

   if -9.8000000000000001e-79 < y.im < 3.20000000000000017e-127

   1. Initial program 70.9%

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

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

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. unpow2 71.9%

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

      4. associate-/r* 82.9%

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

   4. Simplified 82.9%

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

      1. sub-div 85.6%

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

      2. associate-/l* 87.0%

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

   6. Applied egg-rr 87.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 83.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(-1, \frac{y.re}{\frac{y.im}{x.im}}, x.re\right)\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -4.4e+66)
     (* (/ 1.0 (hypot y.re y.im)) (fma -1.0 (/ y.re (/ y.im x.im)) x.re))
     (if (<= y.im -2.4e-78)
       t_0
       (if (<= y.im 1.22e-126)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 1.65e+87)
           t_0
           (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.4e+66) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * fma(-1.0, (y_46_re / (y_46_im / x_46_im)), x_46_re);
	} else if (y_46_im <= -2.4e-78) {
		tmp = t_0;
	} else if (y_46_im <= 1.22e-126) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 1.65e+87) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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)))
	tmp = 0.0
	if (y_46_im <= -4.4e+66)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * fma(-1.0, Float64(y_46_re / Float64(y_46_im / x_46_im)), x_46_re));
	elseif (y_46_im <= -2.4e-78)
		tmp = t_0;
	elseif (y_46_im <= 1.22e-126)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 1.65e+87)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - 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_] := Block[{t$95$0 = 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]}, If[LessEqual[y$46$im, -4.4e+66], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.4e-78], t$95$0, If[LessEqual[y$46$im, 1.22e-126], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+87], t$95$0, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.3999999999999997e66

   1. Initial program 20.0%

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

      1. *-un-lft-identity 20.0%

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

      2. add-sqr-sqrt 20.0%

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

      3. times-frac 20.0%

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

      4. hypot-def 20.0%

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

      5. hypot-def 47.1%

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

   3. Applied egg-rr 47.1%

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

   4. Taylor expanded in y.im around -inf 71.9%

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

      1. fma-def 71.9%

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

      2. associate-/l* 81.1%

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

   6. Simplified 81.1%

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

   if -4.3999999999999997e66 < y.im < -2.4e-78 or 1.21999999999999996e-126 < y.im < 1.6500000000000001e87

   1. Initial program 84.6%

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

   if -2.4e-78 < y.im < 1.21999999999999996e-126

   1. Initial program 70.9%

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

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

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. unpow2 71.9%

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

      4. associate-/r* 82.9%

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

   4. Simplified 82.9%

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

      1. sub-div 85.6%

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

      2. associate-/l* 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 1.6500000000000001e87 < y.im

   1. Initial program 30.5%

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

   2. Taylor expanded in y.re around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. unpow2 75.3%

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

      5. times-frac 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(-1, \frac{y.re}{\frac{y.im}{x.im}}, x.re\right)\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -8.2e+70)
     t_1
     (if (<= y.im -2.7e-78)
       t_0
       (if (<= y.im 1.4e-127)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 5.2e+84) 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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -8.2e+70) {
		tmp = t_1;
	} else if (y_46_im <= -2.7e-78) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-127) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 5.2e+84) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-8.2d+70)) then
        tmp = t_1
    else if (y_46im <= (-2.7d-78)) then
        tmp = t_0
    else if (y_46im <= 1.4d-127) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46im <= 5.2d+84) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -8.2e+70) {
		tmp = t_1;
	} else if (y_46_im <= -2.7e-78) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-127) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 5.2e+84) {
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -8.2e+70:
		tmp = t_1
	elif y_46_im <= -2.7e-78:
		tmp = t_0
	elif y_46_im <= 1.4e-127:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_im <= 5.2e+84:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_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)))
	t_1 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.2e+70)
		tmp = t_1;
	elseif (y_46_im <= -2.7e-78)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-127)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 5.2e+84)
		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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -8.2e+70)
		tmp = t_1;
	elseif (y_46_im <= -2.7e-78)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-127)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_im <= 5.2e+84)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$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]}, Block[{t$95$1 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.2e+70], t$95$1, If[LessEqual[y$46$im, -2.7e-78], t$95$0, If[LessEqual[y$46$im, 1.4e-127], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+84], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.2000000000000004e70 or 5.2000000000000002e84 < y.im

   1. Initial program 25.4%

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

   2. Taylor expanded in y.re around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. mul-1-neg 69.4%

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

      3. unsub-neg 69.4%

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

      4. unpow2 69.4%

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

      5. times-frac 83.3%

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

   4. Simplified 83.3%

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

   if -8.2000000000000004e70 < y.im < -2.69999999999999994e-78 or 1.4e-127 < y.im < 5.2000000000000002e84

   1. Initial program 84.6%

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

   if -2.69999999999999994e-78 < y.im < 1.4e-127

   1. Initial program 70.9%

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

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

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. unpow2 71.9%

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

      4. associate-/r* 82.9%

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

   4. Simplified 82.9%

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

      1. sub-div 85.6%

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

      2. associate-/l* 87.0%

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

   6. Applied egg-rr 87.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91} \lor \neg \left(y.re \leq 21\right):\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.65e+91) (not (<= y.re 21.0)))
   (+ (/ x.im y.re) (/ -1.0 (* (/ y.re y.im) (/ y.re x.re))))
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.65e+91) || !(y_46_re <= 21.0)) {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.65d+91)) .or. (.not. (y_46re <= 21.0d0))) then
        tmp = (x_46im / y_46re) + ((-1.0d0) / ((y_46re / y_46im) * (y_46re / x_46re)))
    else
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.65e+91) || !(y_46_re <= 21.0)) {
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.65e+91) or not (y_46_re <= 21.0):
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)))
	else:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.65e+91) || !(y_46_re <= 21.0))
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(-1.0 / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / x_46_re))));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.65e+91) || ~((y_46_re <= 21.0)))
		tmp = (x_46_im / y_46_re) + (-1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	else
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.65e+91], N[Not[LessEqual[y$46$re, 21.0]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(-1.0 / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.65000000000000009e91 or 21 < y.re

   1. Initial program 46.8%

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

   2. Taylor expanded in y.re around inf 69.8%

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

      1. mul-1-neg 69.8%

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

      2. unsub-neg 69.8%

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

      3. unpow2 69.8%

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

      4. associate-/r* 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.2%

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

      2. inv-pow 73.2%

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

      3. associate-/l* 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. unpow-1 78.4%

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

      2. associate-/r/ 78.4%

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

   8. Simplified 78.4%

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

   if -1.65000000000000009e91 < y.re < 21

   1. Initial program 67.3%

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

   2. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. unpow2 71.6%

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

      5. times-frac 78.7%

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

   4. Simplified 78.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91} \lor \neg \left(y.re \leq 21\right):\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 12.5:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+91)
   (* (- x.im (* x.re (/ y.im y.re))) (/ 1.0 y.re))
   (if (<= y.re 12.5)
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
     (/ (- x.im (/ x.re (/ y.re y.im))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= 12.5) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.15d+91)) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) * (1.0d0 / y_46re)
    else if (y_46re <= 12.5d0) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= 12.5) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.15e+91:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) * (1.0 / y_46_re)
	elif y_46_re <= 12.5:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e+91)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= 12.5)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.15e+91)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) * (1.0 / y_46_re);
	elseif (y_46_re <= 12.5)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e+91], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 12.5], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.14999999999999996e91

   1. Initial program 43.7%

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

   2. Taylor expanded in y.re around inf 65.9%

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

      1. mul-1-neg 65.9%

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

      2. unsub-neg 65.9%

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

      3. unpow2 65.9%

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

      4. associate-/r* 71.4%

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

   4. Simplified 71.4%

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

      1. sub-div 71.4%

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

      2. associate-/l* 74.5%

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

   6. Applied egg-rr 74.5%

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

      1. div-inv 74.5%

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

      2. div-inv 74.5%

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

      3. clear-num 74.5%

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

   8. Applied egg-rr 74.5%

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

   if -1.14999999999999996e91 < y.re < 12.5

   1. Initial program 67.3%

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

   2. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. unpow2 71.6%

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

      5. times-frac 78.7%

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

   4. Simplified 78.7%

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

   if 12.5 < y.re

   1. Initial program 48.5%

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

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

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

      1. mul-1-neg 71.9%

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

      2. unsub-neg 71.9%

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

      3. unpow2 71.9%

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

      4. associate-/r* 74.2%

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

   4. Simplified 74.2%

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

      1. sub-div 74.2%

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

      2. associate-/l* 80.5%

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

   6. Applied egg-rr 80.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 12.5:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 7: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -560000000 \lor \neg \left(y.im \leq 420000\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -560000000.0) (not (<= y.im 420000.0)))
   (- (/ x.re y.im))
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -560000000.0) || !(y_46_im <= 420000.0)) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-560000000.0d0)) .or. (.not. (y_46im <= 420000.0d0))) then
        tmp = -(x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -560000000.0) || !(y_46_im <= 420000.0)) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -560000000.0) or not (y_46_im <= 420000.0):
		tmp = -(x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -560000000.0) || !(y_46_im <= 420000.0))
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -560000000.0) || ~((y_46_im <= 420000.0)))
		tmp = -(x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -560000000.0], N[Not[LessEqual[y$46$im, 420000.0]], $MachinePrecision]], (-N[(x$46$re / y$46$im), $MachinePrecision]), N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.6e8 or 4.2e5 < y.im

   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. Taylor expanded in y.re around 0 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   4. Simplified 66.7%

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

   if -5.6e8 < y.im < 4.2e5

   1. Initial program 79.3%

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

   2. Taylor expanded in y.re around inf 67.5%

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

      1. mul-1-neg 67.5%

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

      2. unsub-neg 67.5%

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

      3. unpow2 67.5%

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

      4. associate-/r* 74.6%

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

   4. Simplified 74.6%

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

      1. sub-div 76.3%

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

      2. associate-/l* 77.0%

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

   6. Applied egg-rr 77.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -560000000 \lor \neg \left(y.im \leq 420000\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 8: 65.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -580000000 \lor \neg \left(y.im \leq 15000000000000\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -580000000.0) (not (<= y.im 15000000000000.0)))
   (- (/ 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 <= -580000000.0) || !(y_46_im <= 15000000000000.0)) {
		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 <= (-580000000.0d0)) .or. (.not. (y_46im <= 15000000000000.0d0))) 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 <= -580000000.0) || !(y_46_im <= 15000000000000.0)) {
		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 <= -580000000.0) or not (y_46_im <= 15000000000000.0):
		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 <= -580000000.0) || !(y_46_im <= 15000000000000.0))
		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 <= -580000000.0) || ~((y_46_im <= 15000000000000.0)))
		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, -580000000.0], N[Not[LessEqual[y$46$im, 15000000000000.0]], $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 < -5.8e8 or 1.5e13 < y.im

   1. Initial program 38.9%

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

   2. Taylor expanded in y.re around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

   if -5.8e8 < y.im < 1.5e13

   1. Initial program 79.1%

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

   2. Taylor expanded in y.re around inf 56.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -580000000 \lor \neg \left(y.im \leq 15000000000000\right):\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.8e+215) (/ 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 <= -6.8e+215) {
		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 <= (-6.8d+215)) 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 <= -6.8e+215) {
		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 <= -6.8e+215:
		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 <= -6.8e+215)
		tmp = 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 <= -6.8e+215)
		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[LessEqual[y$46$im, -6.8e+215], 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 < -6.80000000000000036e215

   1. Initial program 30.3%

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

      1. *-un-lft-identity 30.3%

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

      2. add-sqr-sqrt 30.3%

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

      3. times-frac 30.3%

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

      4. hypot-def 30.3%

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

      5. hypot-def 49.0%

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

   3. Applied egg-rr 49.0%

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

   4. Taylor expanded in y.im around -inf 86.4%

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

      1. fma-def 86.4%

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

      2. associate-/l* 95.5%

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

   6. Simplified 95.5%

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

   7. Taylor expanded in y.re around 0 31.6%

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

   if -6.80000000000000036e215 < y.im

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around inf 41.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 9.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. *-un-lft-identity 59.4%

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

   2. add-sqr-sqrt 59.4%

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

   3. times-frac 59.4%

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

   4. hypot-def 59.4%

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

   5. hypot-def 73.8%

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

3. Applied egg-rr 73.8%

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

4. Taylor expanded in y.im around -inf 28.1%

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

   1. fma-def 28.1%

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

   2. associate-/l* 30.1%

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

6. Simplified 30.1%

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

7. Taylor expanded in y.re around -inf 7.5%

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

8. Final simplification 7.5%

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

Alternative 11: 42.5% 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 59.4%

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

2. Taylor expanded in y.re around inf 38.4%

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

3. Final simplification 38.4%

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

Reproduce

herbie shell --seed 2023238 
(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))))