_divideComplex, imaginary part

Percentage Accurate: 61.9% → 85.2%

Time: 9.1s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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: 85.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \end{array} \end{array} \]

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))) 4e-106)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (- x.re) (/ (pow (hypot y.re y.im) 2.0) 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))) <= 4e-106) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / (pow(hypot(y_46_re, y_46_im), 2.0) / 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))) <= 4e-106)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(-x_46_re) / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / 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[(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], 4e-106], 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[(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] + N[((-x$46$re) / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $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))) < 3.99999999999999976e-106

   1. Initial program 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. add-sqr-sqrt 78.0%

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

      3. times-frac 78.0%

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

      4. hypot-def 78.0%

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

      5. hypot-def 97.7%

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

   3. Applied egg-rr 97.7%

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

   if 3.99999999999999976e-106 < (/.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 49.1%

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

      1. div-sub 45.2%

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

      2. *-commutative 45.2%

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

      3. add-sqr-sqrt 45.2%

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

      4. times-frac 48.8%

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

      5. fma-neg 48.8%

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

      6. hypot-def 48.8%

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

      7. hypot-def 69.2%

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

      8. associate-/l* 75.7%

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

      9. add-sqr-sqrt 75.7%

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

      10. pow2 75.7%

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

      11. hypot-def 75.7%

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

   3. Applied egg-rr 75.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \end{array} \]

Alternative 2: 85.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+305)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (- (* (/ 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);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+305) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (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))) <= 5e+305) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} 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):
	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))) <= 5e+305:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	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(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))) <= 5e+305)
		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(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)
	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))) <= 5e+305)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	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_] := 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], 5e+305], 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[(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 (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.00000000000000009e305

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 5.00000000000000009e305 < (/.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 12.4%

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

   2. Taylor expanded in y.re around 0 42.2%

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

      1. +-commutative 42.2%

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

      2. mul-1-neg 42.2%

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

      3. unsub-neg 42.2%

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

      4. unpow2 42.2%

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

      5. times-frac 57.6%

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

   4. Simplified 57.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

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.re -3.6e+71)
     (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
     (if (<= y.re -1e-141)
       t_0
       (if (<= y.re 1.35e-153)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (if (<= y.re 1.9e+143)
           t_0
           (* (/ y.re (hypot y.re y.im)) (/ x.im (hypot y.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_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -1e-141) {
		tmp = t_0;
	} else if (y_46_re <= 1.35e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.9e+143) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_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)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -1e-141) {
		tmp = t_0;
	} else if (y_46_re <= 1.35e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.9e+143) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_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)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.6e+71:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= -1e-141:
		tmp = t_0
	elif y_46_re <= 1.35e-153:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= 1.9e+143:
		tmp = t_0
	else:
		tmp = (y_46_re / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_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_re <= -3.6e+71)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= -1e-141)
		tmp = t_0;
	elseif (y_46_re <= 1.35e-153)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.9e+143)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_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)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.6e+71)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= -1e-141)
		tmp = t_0;
	elseif (y_46_re <= 1.35e-153)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 1.9e+143)
		tmp = t_0;
	else
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_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[(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$46$re, -3.6e+71], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1e-141], t$95$0, If[LessEqual[y$46$re, 1.35e-153], 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$46$re, 1.9e+143], 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.6e71

   1. Initial program 47.7%

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

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

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. unpow2 75.8%

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

      4. times-frac 87.0%

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

   4. Simplified 87.0%

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

      1. clear-num 87.0%

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

      2. frac-times 88.2%

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

      3. *-un-lft-identity 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/r* 90.1%

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

      2. sub-div 90.0%

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

   8. Applied egg-rr 90.0%

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

   if -3.6e71 < y.re < -1e-141 or 1.35000000000000005e-153 < y.re < 1.9e143

   1. Initial program 81.6%

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

   if -1e-141 < y.re < 1.35000000000000005e-153

   1. Initial program 65.5%

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

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

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

      4. unpow2 82.1%

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

      5. times-frac 90.7%

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

   4. Simplified 90.7%

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

   if 1.9e143 < y.re

   1. Initial program 39.0%

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

   2. Taylor expanded in x.im around inf 39.1%

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

      1. unpow2 39.1%

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

      2. unpow2 39.1%

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

   4. Simplified 39.1%

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

      1. add-sqr-sqrt 39.1%

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

      2. hypot-udef 39.1%

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

      3. hypot-udef 39.1%

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

      4. frac-times 93.4%

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

   6. Applied egg-rr 93.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.66 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \end{array} \end{array} \]

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.re -3.6e+71)
     (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
     (if (<= y.re -1.66e-140)
       t_0
       (if (<= y.re 1.16e-153)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (if (<= y.re 3.45e+142)
           t_0
           (/ x.im (* (hypot y.re y.im) (/ (hypot 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 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_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -1.66e-140) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 3.45e+142) {
		tmp = t_0;
	} else {
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -1.66e-140) {
		tmp = t_0;
	} else if (y_46_re <= 1.16e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 3.45e+142) {
		tmp = t_0;
	} else {
		tmp = x_46_im / (Math.hypot(y_46_re, y_46_im) * (Math.hypot(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):
	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))
	tmp = 0
	if y_46_re <= -3.6e+71:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= -1.66e-140:
		tmp = t_0
	elif y_46_re <= 1.16e-153:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= 3.45e+142:
		tmp = t_0
	else:
		tmp = x_46_im / (math.hypot(y_46_re, y_46_im) * (math.hypot(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)
	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_re <= -3.6e+71)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= -1.66e-140)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-153)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 3.45e+142)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(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)
	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));
	tmp = 0.0;
	if (y_46_re <= -3.6e+71)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= -1.66e-140)
		tmp = t_0;
	elseif (y_46_re <= 1.16e-153)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 3.45e+142)
		tmp = t_0;
	else
		tmp = x_46_im / (hypot(y_46_re, y_46_im) * (hypot(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_] := 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$46$re, -3.6e+71], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.66e-140], t$95$0, If[LessEqual[y$46$re, 1.16e-153], 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$46$re, 3.45e+142], t$95$0, N[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.6e71

   1. Initial program 47.7%

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

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

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. unpow2 75.8%

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

      4. times-frac 87.0%

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

   4. Simplified 87.0%

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

      1. clear-num 87.0%

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

      2. frac-times 88.2%

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

      3. *-un-lft-identity 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/r* 90.1%

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

      2. sub-div 90.0%

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

   8. Applied egg-rr 90.0%

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

   if -3.6e71 < y.re < -1.6600000000000001e-140 or 1.16e-153 < y.re < 3.4500000000000002e142

   1. Initial program 81.6%

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

   if -1.6600000000000001e-140 < y.re < 1.16e-153

   1. Initial program 65.5%

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

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

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

      4. unpow2 82.1%

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

      5. times-frac 90.7%

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

   4. Simplified 90.7%

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

   if 3.4500000000000002e142 < y.re

   1. Initial program 39.0%

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

   2. Taylor expanded in x.im around inf 39.1%

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

      1. unpow2 39.1%

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

      2. unpow2 39.1%

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

   4. Simplified 39.1%

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

      1. add-sqr-sqrt 39.1%

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

      2. hypot-udef 39.1%

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

      3. hypot-udef 39.1%

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

      4. frac-times 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. clear-num 93.5%

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

      2. frac-times 93.5%

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

      3. *-un-lft-identity 93.5%

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

   8. Applied egg-rr 93.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.66 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\\ \end{array} \]

Alternative 5: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3.6e+71)
     (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
     (if (<= y.re -2.25e-142)
       t_0
       (if (<= y.re 1.55e-153)
         (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
         (if (<= y.re 6.8e+58)
           t_0
           (/ (- x.im (* y.im (/ x.re y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -2.25e-142) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.8e+58) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-3.6d+71)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= (-2.25d-142)) then
        tmp = t_0
    else if (y_46re <= 1.55d-153) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= 6.8d+58) then
        tmp = t_0
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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_re <= -3.6e+71) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -2.25e-142) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-153) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 6.8e+58) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.6e+71:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= -2.25e-142:
		tmp = t_0
	elif y_46_re <= 1.55e-153:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= 6.8e+58:
		tmp = t_0
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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_re <= -3.6e+71)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= -2.25e-142)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-153)
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 6.8e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.6e+71)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= -2.25e-142)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-153)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 6.8e+58)
		tmp = t_0;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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$46$re, -3.6e+71], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.25e-142], t$95$0, If[LessEqual[y$46$re, 1.55e-153], 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$46$re, 6.8e+58], t$95$0, N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.6e71

   1. Initial program 47.7%

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

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

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. unpow2 75.8%

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

      4. times-frac 87.0%

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

   4. Simplified 87.0%

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

      1. clear-num 87.0%

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

      2. frac-times 88.2%

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

      3. *-un-lft-identity 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/r* 90.1%

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

      2. sub-div 90.0%

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

   8. Applied egg-rr 90.0%

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

   if -3.6e71 < y.re < -2.25000000000000009e-142 or 1.54999999999999997e-153 < y.re < 6.8000000000000001e58

   1. Initial program 82.1%

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

   if -2.25000000000000009e-142 < y.re < 1.54999999999999997e-153

   1. Initial program 65.5%

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

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

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

      4. unpow2 82.1%

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

      5. times-frac 90.7%

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

   4. Simplified 90.7%

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

   if 6.8000000000000001e58 < y.re

   1. Initial program 51.8%

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

   2. Taylor expanded in y.re around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

      3. unpow2 82.6%

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

      4. times-frac 87.6%

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

   4. Simplified 87.6%

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

      1. associate-*r/ 87.6%

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

      2. sub-div 87.6%

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

   6. Applied egg-rr 87.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.re -6.8e+48)
     (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
     (if (<= y.re -4.8e-91)
       t_0
       (if (<= y.re -1.15e-126)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.re 7e+58) t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -6.8e+48) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -4.8e-91) {
		tmp = t_0;
	} else if (y_46_re <= -1.15e-126) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 7e+58) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46re <= (-6.8d+48)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= (-4.8d-91)) then
        tmp = t_0
    else if (y_46re <= (-1.15d-126)) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else if (y_46re <= 7d+58) then
        tmp = t_0
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -6.8e+48) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -4.8e-91) {
		tmp = t_0;
	} else if (y_46_re <= -1.15e-126) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 7e+58) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_re <= -6.8e+48:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= -4.8e-91:
		tmp = t_0
	elif y_46_re <= -1.15e-126:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	elif y_46_re <= 7e+58:
		tmp = t_0
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_re <= -6.8e+48)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= -4.8e-91)
		tmp = t_0;
	elseif (y_46_re <= -1.15e-126)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= 7e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -6.8e+48)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= -4.8e-91)
		tmp = t_0;
	elseif (y_46_re <= -1.15e-126)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_re <= 7e+58)
		tmp = t_0;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / 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$46$re, -6.8e+48], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.8e-91], t$95$0, If[LessEqual[y$46$re, -1.15e-126], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 7e+58], t$95$0, N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.8000000000000006e48

   1. Initial program 53.4%

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

   2. Taylor expanded in y.re around inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. unsub-neg 75.2%

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

      3. unpow2 75.2%

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

      4. times-frac 85.1%

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

   4. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. frac-times 86.2%

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

      3. *-un-lft-identity 86.2%

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

   6. Applied egg-rr 86.2%

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

      1. associate-/r* 87.8%

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

      2. sub-div 87.8%

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

   8. Applied egg-rr 87.8%

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

   if -6.8000000000000006e48 < y.re < -4.80000000000000022e-91 or -1.15000000000000005e-126 < y.re < 6.9999999999999995e58

   1. Initial program 72.4%

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

   2. Taylor expanded in y.re around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 79.2%

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

   4. Simplified 79.2%

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

   if -4.80000000000000022e-91 < y.re < -1.15000000000000005e-126

   1. Initial program 99.6%

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

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

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

      1. mul-1-neg 86.4%

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

      2. unsub-neg 86.4%

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

      3. unpow2 86.4%

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

      4. times-frac 72.1%

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

   4. Simplified 72.1%

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

      1. associate-*r/ 72.1%

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

      2. sub-div 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in x.re around 0 86.4%

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

   if 6.9999999999999995e58 < y.re

   1. Initial program 53.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. unsub-neg 84.4%

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

      3. unpow2 84.4%

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

      4. times-frac 89.6%

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

   4. Simplified 89.6%

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

      1. associate-*r/ 89.6%

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

      2. sub-div 89.6%

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

   6. Applied egg-rr 89.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+58}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48} \lor \neg \left(y.re \leq -4.8 \cdot 10^{-30}\right) \land \left(y.re \leq -5 \cdot 10^{-148} \lor \neg \left(y.re \leq 5.2 \cdot 10^{+25}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.8e+48)
         (and (not (<= y.re -4.8e-30))
              (or (<= y.re -5e-148) (not (<= y.re 5.2e+25)))))
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (- (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.8e+48) || (!(y_46_re <= -4.8e-30) && ((y_46_re <= -5e-148) || !(y_46_re <= 5.2e+25)))) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6.8d+48)) .or. (.not. (y_46re <= (-4.8d-30))) .and. (y_46re <= (-5d-148)) .or. (.not. (y_46re <= 5.2d+25))) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = -(x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.8e+48) || (!(y_46_re <= -4.8e-30) && ((y_46_re <= -5e-148) || !(y_46_re <= 5.2e+25)))) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = -(x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6.8e+48) or (not (y_46_re <= -4.8e-30) and ((y_46_re <= -5e-148) or not (y_46_re <= 5.2e+25))):
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = -(x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.8e+48) || (!(y_46_re <= -4.8e-30) && ((y_46_re <= -5e-148) || !(y_46_re <= 5.2e+25))))
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = Float64(-Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.8e+48) || (~((y_46_re <= -4.8e-30)) && ((y_46_re <= -5e-148) || ~((y_46_re <= 5.2e+25)))))
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = -(x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6.8e+48], And[N[Not[LessEqual[y$46$re, -4.8e-30]], $MachinePrecision], Or[LessEqual[y$46$re, -5e-148], N[Not[LessEqual[y$46$re, 5.2e+25]], $MachinePrecision]]]], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], (-N[(x$46$re / y$46$im), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.8000000000000006e48 or -4.7999999999999997e-30 < y.re < -4.9999999999999999e-148 or 5.1999999999999997e25 < y.re

   1. Initial program 62.2%

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

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

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

      3. unpow2 74.3%

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

      4. times-frac 79.3%

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

   4. Simplified 79.3%

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

      1. associate-*r/ 80.4%

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

      2. sub-div 80.4%

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

   6. Applied egg-rr 80.4%

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

   if -6.8000000000000006e48 < y.re < -4.7999999999999997e-30 or -4.9999999999999999e-148 < y.re < 5.1999999999999997e25

   1. Initial program 69.8%

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

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

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   4. Simplified 68.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48} \lor \neg \left(y.re \leq -4.8 \cdot 10^{-30}\right) \land \left(y.re \leq -5 \cdot 10^{-148} \lor \neg \left(y.re \leq 5.2 \cdot 10^{+25}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-28} \lor \neg \left(y.re \leq -5 \cdot 10^{-148}\right) \land y.re \leq 4.6 \cdot 10^{+25}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.8e+48)
   (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
   (if (or (<= y.re -6.8e-28) (and (not (<= y.re -5e-148)) (<= y.re 4.6e+25)))
     (- (/ x.re y.im))
     (/ (- x.im (* y.im (/ x.re y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.8e+48) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if ((y_46_re <= -6.8e-28) || (!(y_46_re <= -5e-148) && (y_46_re <= 4.6e+25))) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-6.8d+48)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if ((y_46re <= (-6.8d-28)) .or. (.not. (y_46re <= (-5d-148))) .and. (y_46re <= 4.6d+25)) then
        tmp = -(x_46re / y_46im)
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -6.8e+48) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if ((y_46_re <= -6.8e-28) || (!(y_46_re <= -5e-148) && (y_46_re <= 4.6e+25))) {
		tmp = -(x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -6.8e+48:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif (y_46_re <= -6.8e-28) or (not (y_46_re <= -5e-148) and (y_46_re <= 4.6e+25)):
		tmp = -(x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -6.8e+48)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif ((y_46_re <= -6.8e-28) || (!(y_46_re <= -5e-148) && (y_46_re <= 4.6e+25)))
		tmp = Float64(-Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -6.8e+48)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif ((y_46_re <= -6.8e-28) || (~((y_46_re <= -5e-148)) && (y_46_re <= 4.6e+25)))
		tmp = -(x_46_re / y_46_im);
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.8e+48], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, -6.8e-28], And[N[Not[LessEqual[y$46$re, -5e-148]], $MachinePrecision], LessEqual[y$46$re, 4.6e+25]]], (-N[(x$46$re / y$46$im), $MachinePrecision]), N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.8000000000000006e48

   1. Initial program 53.4%

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

   2. Taylor expanded in y.re around inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. unsub-neg 75.2%

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

      3. unpow2 75.2%

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

      4. times-frac 85.1%

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

   4. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. frac-times 86.2%

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

      3. *-un-lft-identity 86.2%

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

   6. Applied egg-rr 86.2%

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

      1. associate-/r* 87.8%

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

      2. sub-div 87.8%

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

   8. Applied egg-rr 87.8%

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

   if -6.8000000000000006e48 < y.re < -6.8000000000000001e-28 or -4.9999999999999999e-148 < y.re < 4.5999999999999996e25

   1. Initial program 69.8%

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

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

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   4. Simplified 68.5%

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

   if -6.8000000000000001e-28 < y.re < -4.9999999999999999e-148 or 4.5999999999999996e25 < y.re

   1. Initial program 68.5%

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

   2. Taylor expanded in y.re around inf 73.7%

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

      1. mul-1-neg 73.7%

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

      2. unsub-neg 73.7%

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

      3. unpow2 73.7%

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

      4. times-frac 75.1%

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

   4. Simplified 75.1%

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

      1. associate-*r/ 75.1%

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

      2. sub-div 75.1%

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

   6. Applied egg-rr 75.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-28} \lor \neg \left(y.re \leq -5 \cdot 10^{-148}\right) \land y.re \leq 4.6 \cdot 10^{+25}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.re y.im))))
   (if (<= y.re -1.25e+49)
     (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
     (if (<= y.re -4e-32)
       t_0
       (if (<= y.re -5e-148)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.re 2e+26) t_0 (/ (- x.im (* y.im (/ x.re y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -(x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -1.25e+49) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -4e-32) {
		tmp = t_0;
	} else if (y_46_re <= -5e-148) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 2e+26) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x_46re / y_46im)
    if (y_46re <= (-1.25d+49)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= (-4d-32)) then
        tmp = t_0
    else if (y_46re <= (-5d-148)) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else if (y_46re <= 2d+26) then
        tmp = t_0
    else
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -(x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -1.25e+49) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= -4e-32) {
		tmp = t_0;
	} else if (y_46_re <= -5e-148) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_re <= 2e+26) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -(x_46_re / y_46_im)
	tmp = 0
	if y_46_re <= -1.25e+49:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= -4e-32:
		tmp = t_0
	elif y_46_re <= -5e-148:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	elif y_46_re <= 2e+26:
		tmp = t_0
	else:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(-Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.25e+49)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= -4e-32)
		tmp = t_0;
	elseif (y_46_re <= -5e-148)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_re <= 2e+26)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -(x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.25e+49)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= -4e-32)
		tmp = t_0;
	elseif (y_46_re <= -5e-148)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_re <= 2e+26)
		tmp = t_0;
	else
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = (-N[(x$46$re / y$46$im), $MachinePrecision])}, If[LessEqual[y$46$re, -1.25e+49], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4e-32], t$95$0, If[LessEqual[y$46$re, -5e-148], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2e+26], t$95$0, N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.2500000000000001e49

   1. Initial program 53.4%

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

   2. Taylor expanded in y.re around inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. unsub-neg 75.2%

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

      3. unpow2 75.2%

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

      4. times-frac 85.1%

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

   4. Simplified 85.1%

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

      1. clear-num 85.1%

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

      2. frac-times 86.2%

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

      3. *-un-lft-identity 86.2%

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

   6. Applied egg-rr 86.2%

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

      1. associate-/r* 87.8%

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

      2. sub-div 87.8%

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

   8. Applied egg-rr 87.8%

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

   if -1.2500000000000001e49 < y.re < -4.00000000000000022e-32 or -4.9999999999999999e-148 < y.re < 2.0000000000000001e26

   1. Initial program 69.8%

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

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

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   4. Simplified 68.5%

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

   if -4.00000000000000022e-32 < y.re < -4.9999999999999999e-148

   1. Initial program 96.0%

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

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

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

      1. mul-1-neg 62.5%

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

      2. unsub-neg 62.5%

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

      3. unpow2 62.5%

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

      4. times-frac 58.3%

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

   4. Simplified 58.3%

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

      1. associate-*r/ 58.4%

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

      2. sub-div 58.4%

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

   6. Applied egg-rr 58.4%

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

   7. Taylor expanded in x.re around 0 62.5%

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

   if 2.0000000000000001e26 < y.re

   1. Initial program 54.7%

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

   2. Taylor expanded in y.re around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. unpow2 79.2%

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

      4. times-frac 83.5%

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

   4. Simplified 83.5%

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

      1. associate-*r/ 83.5%

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

      2. sub-div 83.5%

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

   6. Applied egg-rr 83.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-32}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+26}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 10: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-91} \lor \neg \left(y.re \leq -6.2 \cdot 10^{-143}\right) \land y.re \leq 1.95 \cdot 10^{+60}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.4e+49)
   (/ x.im y.re)
   (if (or (<= y.re -4.8e-91)
           (and (not (<= y.re -6.2e-143)) (<= y.re 1.95e+60)))
     (- (/ 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_re <= -4.4e+49) {
		tmp = x_46_im / y_46_re;
	} else if ((y_46_re <= -4.8e-91) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 1.95e+60))) {
		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_46re <= (-4.4d+49)) then
        tmp = x_46im / y_46re
    else if ((y_46re <= (-4.8d-91)) .or. (.not. (y_46re <= (-6.2d-143))) .and. (y_46re <= 1.95d+60)) 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_re <= -4.4e+49) {
		tmp = x_46_im / y_46_re;
	} else if ((y_46_re <= -4.8e-91) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 1.95e+60))) {
		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_re <= -4.4e+49:
		tmp = x_46_im / y_46_re
	elif (y_46_re <= -4.8e-91) or (not (y_46_re <= -6.2e-143) and (y_46_re <= 1.95e+60)):
		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_re <= -4.4e+49)
		tmp = Float64(x_46_im / y_46_re);
	elseif ((y_46_re <= -4.8e-91) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 1.95e+60)))
		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_re <= -4.4e+49)
		tmp = x_46_im / y_46_re;
	elseif ((y_46_re <= -4.8e-91) || (~((y_46_re <= -6.2e-143)) && (y_46_re <= 1.95e+60)))
		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$46$re, -4.4e+49], N[(x$46$im / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, -4.8e-91], And[N[Not[LessEqual[y$46$re, -6.2e-143]], $MachinePrecision], LessEqual[y$46$re, 1.95e+60]]], (-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.re < -4.4000000000000001e49 or -4.80000000000000022e-91 < y.re < -6.20000000000000015e-143 or 1.95000000000000015e60 < y.re

   1. Initial program 58.8%

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

   2. Taylor expanded in y.re around inf 70.2%

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

   if -4.4000000000000001e49 < y.re < -4.80000000000000022e-91 or -6.20000000000000015e-143 < y.re < 1.95000000000000015e60

   1. Initial program 71.2%

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

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

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-91} \lor \neg \left(y.re \leq -6.2 \cdot 10^{-143}\right) \land y.re \leq 1.95 \cdot 10^{+60}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11: 42.6% 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 65.9%

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

2. Taylor expanded in y.re around inf 38.7%

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

3. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023174 
(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))))