_divideComplex, imaginary part

Percentage Accurate: 61.4% → 96.8%

Time: 12.1s

Alternatives: 19

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 96.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\\ {t_0}^{2} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)} \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (cbrt
          (-
           (/ y.re (/ (hypot y.re y.im) x.im))
           (* (/ y.im (hypot y.re y.im)) x.re)))))
   (* (pow t_0 2.0) (/ t_0 (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 = cbrt(((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - ((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re)));
	return pow(t_0, 2.0) * (t_0 / hypot(y_46_re, y_46_im));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.cbrt(((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - ((y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re)));
	return Math.pow(t_0, 2.0) * (t_0 / Math.hypot(y_46_re, y_46_im));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cbrt(Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * x_46_re)))
	return Float64((t_0 ^ 2.0) * Float64(t_0 / hypot(y_46_re, y_46_im)))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.1%

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

   1. *-un-lft-identity 63.1%

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

   2. add-sqr-sqrt 63.1%

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

   3. times-frac 63.1%

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

   4. hypot-def 63.1%

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

   5. hypot-def 75.8%

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

3. Applied egg-rr 75.8%

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

   1. associate-*l/ 76.0%

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

   2. *-un-lft-identity 76.0%

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

   3. *-commutative 76.0%

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

   4. *-commutative 76.0%

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

5. Applied egg-rr 76.0%

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

   1. div-sub 76.0%

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

7. Applied egg-rr 76.0%

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

   1. associate-/l* 86.3%

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

9. Simplified 86.3%

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

    1. add-cube-cbrt 85.2%

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}} \cdot \sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right) \cdot \sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

    2. *-un-lft-identity 85.2%

       \[\leadsto \frac{\left(\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}} \cdot \sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right) \cdot \sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\color{blue}{1 \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \]

    3. times-frac 85.2%

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}} \cdot \sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{1} \cdot \frac{\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

    4. pow2 85.2%

       \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

    5. associate-/l* 83.0%

       \[\leadsto \frac{{\left(\sqrt[3]{\color{blue}{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

    6. associate-/r/ 83.5%

       \[\leadsto \frac{{\left(\sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

11. Applied egg-rr 95.8%

    \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

12. Final simplification 95.8%

    \[\leadsto {\left(\sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right)}^{2} \cdot \frac{\sqrt[3]{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

Alternative 2: 92.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (/ (hypot y.re y.im) x.re))))
   (if (<= y.re -4.7e+126)
     (/ (- (- x.im) t_0) (hypot y.re y.im))
     (if (<= y.re 2.5e+126)
       (/ (- (/ (* y.re x.im) (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 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -4.7e+126) {
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 2.5e+126) {
		tmp = (((y_46_re * x_46_im) / 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(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.7e+126)
		tmp = Float64(Float64(Float64(-x_46_im) - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 2.5e+126)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / hypot(y_46_re, y_46_im)) - 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[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.7e+126], N[(N[((-x$46$im) - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+126], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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 3 regimes

2. if y.re < -4.6999999999999999e126

   1. Initial program 21.4%

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

      1. *-un-lft-identity 21.4%

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

      2. add-sqr-sqrt 21.4%

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

      3. times-frac 21.4%

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

      4. hypot-def 21.4%

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

      5. hypot-def 52.4%

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

   3. Applied egg-rr 52.4%

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

      1. associate-*l/ 52.6%

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

      2. *-un-lft-identity 52.6%

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

      3. *-commutative 52.6%

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

      4. *-commutative 52.6%

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

   5. Applied egg-rr 52.6%

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

      1. div-sub 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-/l* 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in y.re around -inf 92.2%

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

       1. neg-mul-1 92.2%

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

   12. Simplified 92.2%

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

   if -4.6999999999999999e126 < y.re < 2.49999999999999989e126

   1. Initial program 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. add-sqr-sqrt 76.1%

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

      3. times-frac 76.0%

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

      4. hypot-def 76.0%

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

      5. hypot-def 84.0%

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

   3. Applied egg-rr 84.0%

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

      1. associate-*l/ 84.3%

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

      2. *-un-lft-identity 84.3%

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

      3. *-commutative 84.3%

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

      4. *-commutative 84.3%

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

   5. Applied egg-rr 84.3%

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

      1. div-sub 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   if 2.49999999999999989e126 < y.re

   1. Initial program 42.8%

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

      1. div-sub 42.8%

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

      2. *-commutative 42.8%

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

      3. add-sqr-sqrt 42.8%

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

      4. times-frac 46.6%

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

      5. fma-neg 46.6%

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

      6. hypot-def 46.6%

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

      7. hypot-def 92.4%

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

      8. associate-/l* 92.8%

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

      9. add-sqr-sqrt 92.8%

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

      10. pow2 92.8%

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

      11. hypot-def 92.8%

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

   3. Applied egg-rr 92.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \end{array} \]

Alternative 3: 88.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+298)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (- x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+298) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / 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 / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+298)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / 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) / 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[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $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+298], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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) / 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))) < 5.0000000000000003e298

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 79.9%

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

      4. hypot-def 79.9%

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

      5. hypot-def 95.1%

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

   3. Applied egg-rr 95.1%

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

      1. associate-*l/ 95.3%

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

      2. *-un-lft-identity 95.3%

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

      3. *-commutative 95.3%

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

      4. *-commutative 95.3%

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

   5. Applied egg-rr 95.3%

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

   if 5.0000000000000003e298 < (/.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 11.4%

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

      1. div-sub 8.1%

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

      2. *-commutative 8.1%

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

      3. add-sqr-sqrt 8.1%

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

      4. times-frac 7.9%

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

      5. fma-neg 7.9%

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

      6. hypot-def 7.9%

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

      7. hypot-def 44.6%

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

      8. associate-/l* 52.4%

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

      9. add-sqr-sqrt 52.4%

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

      10. pow2 52.3%

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

      11. hypot-def 52.3%

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

   3. Applied egg-rr 52.3%

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

   4. Taylor expanded in y.re around 0 72.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternative 4: 94.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (/ (hypot y.re y.im) x.re))))
   (if (<= y.re -7.2e+125)
     (/ (- (- x.im) t_0) (hypot y.re y.im))
     (if (<= y.re 5.8e+99)
       (/ (- (/ (* y.re x.im) (hypot y.re y.im)) t_0) (hypot y.re y.im))
       (/ (- x.im t_0) (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 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+125) {
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 5.8e+99) {
		tmp = (((y_46_re * x_46_im) / hypot(y_46_re, y_46_im)) - t_0) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im - t_0) / 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 = y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+125) {
		tmp = (-x_46_im - t_0) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 5.8e+99) {
		tmp = (((y_46_re * x_46_im) / Math.hypot(y_46_re, y_46_im)) - t_0) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im - t_0) / 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 = y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)
	tmp = 0
	if y_46_re <= -7.2e+125:
		tmp = (-x_46_im - t_0) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= 5.8e+99:
		tmp = (((y_46_re * x_46_im) / math.hypot(y_46_re, y_46_im)) - t_0) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (x_46_im - t_0) / 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(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.2e+125)
		tmp = Float64(Float64(Float64(-x_46_im) - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 5.8e+99)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / hypot(y_46_re, y_46_im)) - t_0) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im - t_0) / 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 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7.2e+125)
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= 5.8e+99)
		tmp = (((y_46_re * x_46_im) / hypot(y_46_re, y_46_im)) - t_0) / hypot(y_46_re, y_46_im);
	else
		tmp = (x_46_im - t_0) / 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[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e+125], N[(N[((-x$46$im) - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+99], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.2000000000000007e125

   1. Initial program 21.4%

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

      1. *-un-lft-identity 21.4%

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

      2. add-sqr-sqrt 21.4%

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

      3. times-frac 21.4%

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

      4. hypot-def 21.4%

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

      5. hypot-def 52.4%

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

   3. Applied egg-rr 52.4%

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

      1. associate-*l/ 52.6%

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

      2. *-un-lft-identity 52.6%

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

      3. *-commutative 52.6%

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

      4. *-commutative 52.6%

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

   5. Applied egg-rr 52.6%

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

      1. div-sub 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-/l* 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in y.re around -inf 92.2%

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

       1. neg-mul-1 92.2%

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

   12. Simplified 92.2%

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

   if -7.2000000000000007e125 < y.re < 5.8000000000000004e99

   1. Initial program 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. add-sqr-sqrt 75.7%

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

      3. times-frac 75.6%

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

      4. hypot-def 75.6%

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

      5. hypot-def 83.4%

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

   3. Applied egg-rr 83.4%

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

      1. associate-*l/ 83.6%

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

      2. *-un-lft-identity 83.6%

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

      3. *-commutative 83.6%

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

      4. *-commutative 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. div-sub 83.6%

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

   7. Applied egg-rr 83.6%

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

      1. associate-/l* 96.6%

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

   9. Simplified 96.6%

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

   if 5.8000000000000004e99 < y.re

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.3%

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

      4. hypot-def 49.3%

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

      5. hypot-def 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. associate-*l/ 66.2%

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

      2. *-un-lft-identity 66.2%

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

      3. *-commutative 66.2%

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

      4. *-commutative 66.2%

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

   5. Applied egg-rr 66.2%

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

      1. div-sub 66.2%

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

   7. Applied egg-rr 66.2%

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

      1. associate-/l* 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in y.re around inf 90.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 89.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.05 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im))
          (hypot y.re y.im)))
        (t_1 (/ y.im (/ (hypot y.re y.im) x.re))))
   (if (<= y.re -2.2e+63)
     (/ (- (- x.im) t_1) (hypot y.re y.im))
     (if (<= y.re -8e-140)
       t_0
       (if (<= y.re 3.05e-155)
         (* (/ -1.0 y.im) (+ x.re (/ -1.0 (/ (/ y.im y.re) x.im))))
         (if (<= y.re 3.4e+95) t_0 (/ (- x.im t_1) (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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double t_1 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -2.2e+63) {
		tmp = (-x_46_im - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -8e-140) {
		tmp = t_0;
	} else if (y_46_re <= 3.05e-155) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else if (y_46_re <= 3.4e+95) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	double t_1 = y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -2.2e+63) {
		tmp = (-x_46_im - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -8e-140) {
		tmp = t_0;
	} else if (y_46_re <= 3.05e-155) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else if (y_46_re <= 3.4e+95) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / 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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	t_1 = y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)
	tmp = 0
	if y_46_re <= -2.2e+63:
		tmp = (-x_46_im - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -8e-140:
		tmp = t_0
	elif y_46_re <= 3.05e-155:
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)))
	elif y_46_re <= 3.4e+95:
		tmp = t_0
	else:
		tmp = (x_46_im - t_1) / 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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	t_1 = Float64(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))
	tmp = 0.0
	if (y_46_re <= -2.2e+63)
		tmp = Float64(Float64(Float64(-x_46_im) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -8e-140)
		tmp = t_0;
	elseif (y_46_re <= 3.05e-155)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re + Float64(-1.0 / Float64(Float64(y_46_im / y_46_re) / x_46_im))));
	elseif (y_46_re <= 3.4e+95)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - t_1) / 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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	t_1 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.2e+63)
		tmp = (-x_46_im - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -8e-140)
		tmp = t_0;
	elseif (y_46_re <= 3.05e-155)
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	elseif (y_46_re <= 3.4e+95)
		tmp = t_0;
	else
		tmp = (x_46_im - t_1) / 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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+63], N[(N[((-x$46$im) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8e-140], t$95$0, If[LessEqual[y$46$re, 3.05e-155], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re + N[(-1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.4e+95], t$95$0, N[(N[(x$46$im - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1999999999999999e63

   1. Initial program 28.1%

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

      1. *-un-lft-identity 28.1%

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

      2. add-sqr-sqrt 28.1%

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

      3. times-frac 28.1%

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

      4. hypot-def 28.1%

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

      5. hypot-def 53.2%

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

   3. Applied egg-rr 53.2%

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

      1. associate-*l/ 53.4%

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

      2. *-un-lft-identity 53.4%

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

      3. *-commutative 53.4%

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

      4. *-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. div-sub 53.4%

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

   7. Applied egg-rr 53.4%

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

      1. associate-/l* 68.2%

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

   9. Simplified 68.2%

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

   10. Taylor expanded in y.re around -inf 93.7%

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

       1. neg-mul-1 93.7%

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

   12. Simplified 93.7%

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

   if -2.1999999999999999e63 < y.re < -7.9999999999999999e-140 or 3.05000000000000012e-155 < y.re < 3.40000000000000022e95

   1. Initial program 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. add-sqr-sqrt 84.0%

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

      3. times-frac 83.9%

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

      4. hypot-def 83.9%

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

      5. hypot-def 92.1%

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

   3. Applied egg-rr 92.1%

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

      1. associate-*l/ 92.3%

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

      2. *-un-lft-identity 92.3%

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

      3. *-commutative 92.3%

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

      4. *-commutative 92.3%

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

   5. Applied egg-rr 92.3%

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

   if -7.9999999999999999e-140 < y.re < 3.05000000000000012e-155

   1. Initial program 67.8%

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

      1. *-un-lft-identity 67.8%

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

      2. add-sqr-sqrt 67.8%

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

      3. times-frac 67.8%

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

      4. hypot-def 67.8%

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

      5. hypot-def 75.9%

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

   3. Applied egg-rr 75.9%

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

   4. Taylor expanded in y.im around -inf 49.7%

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

   5. Taylor expanded in y.im around -inf 97.3%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

   7. Applied egg-rr 97.2%

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

      1. unpow-1 97.2%

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

      2. associate-/r* 98.5%

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

   9. Simplified 98.5%

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

   if 3.40000000000000022e95 < y.re

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.3%

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

      4. hypot-def 49.3%

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

      5. hypot-def 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. associate-*l/ 66.2%

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

      2. *-un-lft-identity 66.2%

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

      3. *-commutative 66.2%

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

      4. *-commutative 66.2%

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

   5. Applied egg-rr 66.2%

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

      1. div-sub 66.2%

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

   7. Applied egg-rr 66.2%

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

      1. associate-/l* 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in y.re around inf 90.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.05 \cdot 10^{-155}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 84.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.55e+96)
   (- (/ x.im y.re) (/ 1.0 (* (/ y.re y.im) (/ y.re x.re))))
   (if (<= y.re -3.9e-140)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.65e-94)
       (* (/ -1.0 y.im) (+ x.re (/ -1.0 (/ (/ y.im y.re) x.im))))
       (/ (- x.im (/ y.im (/ (hypot y.re y.im) x.re))) (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 tmp;
	if (y_46_re <= -1.55e+96) {
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -3.9e-140) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.65e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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 tmp;
	if (y_46_re <= -1.55e+96) {
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -3.9e-140) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.65e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - (y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re))) / 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):
	tmp = 0
	if y_46_re <= -1.55e+96:
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)))
	elif y_46_re <= -3.9e-140:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.65e-94:
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)))
	else:
		tmp = (x_46_im - (y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re))) / 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)
	tmp = 0.0
	if (y_46_re <= -1.55e+96)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(1.0 / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / x_46_re))));
	elseif (y_46_re <= -3.9e-140)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.65e-94)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re + Float64(-1.0 / Float64(Float64(y_46_im / y_46_re) / x_46_im))));
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))) / 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)
	tmp = 0.0;
	if (y_46_re <= -1.55e+96)
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	elseif (y_46_re <= -3.9e-140)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.65e-94)
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	else
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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_] := If[LessEqual[y$46$re, -1.55e+96], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(1.0 / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.9e-140], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e-94], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re + N[(-1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.5499999999999999e96

   1. Initial program 26.5%

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

   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 84.9%

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

   4. Simplified 84.9%

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

      1. frac-times 73.7%

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

      2. *-commutative 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. clear-num 73.7%

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

      2. inv-pow 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. unpow-1 73.7%

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

      2. times-frac 85.0%

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

   10. Simplified 85.0%

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

   if -1.5499999999999999e96 < y.re < -3.90000000000000019e-140

   1. Initial program 83.5%

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

   if -3.90000000000000019e-140 < y.re < 1.6500000000000001e-94

   1. Initial program 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. add-sqr-sqrt 71.6%

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

      3. times-frac 71.5%

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

      4. hypot-def 71.5%

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

      5. hypot-def 79.5%

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

   3. Applied egg-rr 79.5%

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

   4. Taylor expanded in y.im around -inf 47.9%

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

   5. Taylor expanded in y.im around -inf 93.9%

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

      1. clear-num 93.9%

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

      2. inv-pow 93.9%

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

   7. Applied egg-rr 93.9%

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

      1. unpow-1 93.9%

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

      2. associate-/r* 95.0%

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

   9. Simplified 95.0%

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

   if 1.6500000000000001e-94 < 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. Step-by-step derivation

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.2%

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

      3. times-frac 62.1%

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

      4. hypot-def 62.1%

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

      5. hypot-def 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. associate-*l/ 76.5%

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

      2. *-un-lft-identity 76.5%

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

      3. *-commutative 76.5%

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

      4. *-commutative 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. div-sub 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/l* 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in y.re around inf 87.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+96}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-140}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 7: 86.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.52 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (/ (hypot y.re y.im) x.re))))
   (if (<= y.re -9.5e+63)
     (/ (- (- x.im) t_0) (hypot y.re y.im))
     (if (<= y.re -1.52e-139)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.9e-94)
         (* (/ -1.0 y.im) (+ x.re (/ -1.0 (/ (/ y.im y.re) x.im))))
         (/ (- x.im t_0) (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 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -9.5e+63) {
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.52e-139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.9e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - t_0) / 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 = y_46_im / (Math.hypot(y_46_re, y_46_im) / x_46_re);
	double tmp;
	if (y_46_re <= -9.5e+63) {
		tmp = (-x_46_im - t_0) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.52e-139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.9e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - t_0) / 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 = y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re)
	tmp = 0
	if y_46_re <= -9.5e+63:
		tmp = (-x_46_im - t_0) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -1.52e-139:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.9e-94:
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)))
	else:
		tmp = (x_46_im - t_0) / 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(y_46_im / Float64(hypot(y_46_re, y_46_im) / x_46_re))
	tmp = 0.0
	if (y_46_re <= -9.5e+63)
		tmp = Float64(Float64(Float64(-x_46_im) - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.52e-139)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.9e-94)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re + Float64(-1.0 / Float64(Float64(y_46_im / y_46_re) / x_46_im))));
	else
		tmp = Float64(Float64(x_46_im - t_0) / 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 = y_46_im / (hypot(y_46_re, y_46_im) / x_46_re);
	tmp = 0.0;
	if (y_46_re <= -9.5e+63)
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -1.52e-139)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.9e-94)
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	else
		tmp = (x_46_im - t_0) / 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[(y$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e+63], N[(N[((-x$46$im) - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.52e-139], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-94], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re + N[(-1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.5000000000000003e63

   1. Initial program 28.1%

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

      1. *-un-lft-identity 28.1%

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

      2. add-sqr-sqrt 28.1%

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

      3. times-frac 28.1%

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

      4. hypot-def 28.1%

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

      5. hypot-def 53.2%

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

   3. Applied egg-rr 53.2%

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

      1. associate-*l/ 53.4%

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

      2. *-un-lft-identity 53.4%

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

      3. *-commutative 53.4%

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

      4. *-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. div-sub 53.4%

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

   7. Applied egg-rr 53.4%

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

      1. associate-/l* 68.2%

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

   9. Simplified 68.2%

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

   10. Taylor expanded in y.re around -inf 93.7%

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

       1. neg-mul-1 93.7%

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

   12. Simplified 93.7%

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

   if -9.5000000000000003e63 < y.re < -1.51999999999999999e-139

   1. Initial program 89.2%

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

   if -1.51999999999999999e-139 < y.re < 1.9e-94

   1. Initial program 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. add-sqr-sqrt 71.6%

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

      3. times-frac 71.5%

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

      4. hypot-def 71.5%

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

      5. hypot-def 79.5%

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

   3. Applied egg-rr 79.5%

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

   4. Taylor expanded in y.im around -inf 47.9%

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

   5. Taylor expanded in y.im around -inf 93.9%

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

      1. clear-num 93.9%

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

      2. inv-pow 93.9%

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

   7. Applied egg-rr 93.9%

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

      1. unpow-1 93.9%

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

      2. associate-/r* 95.0%

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

   9. Simplified 95.0%

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

   if 1.9e-94 < 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. Step-by-step derivation

      1. *-un-lft-identity 62.2%

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

      2. add-sqr-sqrt 62.2%

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

      3. times-frac 62.1%

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

      4. hypot-def 62.1%

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

      5. hypot-def 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. associate-*l/ 76.5%

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

      2. *-un-lft-identity 76.5%

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

      3. *-commutative 76.5%

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

      4. *-commutative 76.5%

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

   5. Applied egg-rr 76.5%

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

      1. div-sub 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/l* 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in y.re around inf 87.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.52 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 8: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -2.2e+95)
     (- (/ x.im y.re) (/ 1.0 (* (/ y.re y.im) (/ y.re x.re))))
     (if (<= y.re -1.6e-139)
       (/ (- (* y.re x.im) (* y.im x.re)) t_0)
       (if (<= y.re 2e-151)
         (* (/ -1.0 y.im) (+ x.re (/ -1.0 (/ (/ y.im y.re) x.im))))
         (if (<= y.re 4.8e+96)
           (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -2.2e+95) {
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -1.6e-139) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / t_0;
	} else if (y_46_re <= 2e-151) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else if (y_46_re <= 4.8e+96) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.2e+95)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(1.0 / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / x_46_re))));
	elseif (y_46_re <= -1.6e-139)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / t_0);
	elseif (y_46_re <= 2e-151)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re + Float64(-1.0 / Float64(Float64(y_46_im / y_46_re) / x_46_im))));
	elseif (y_46_re <= 4.8e+96)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+95], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(1.0 / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.6e-139], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2e-151], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re + N[(-1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.8e+96], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.1999999999999999e95

   1. Initial program 26.5%

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

   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 84.9%

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

   4. Simplified 84.9%

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

      1. frac-times 73.7%

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

      2. *-commutative 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. clear-num 73.7%

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

      2. inv-pow 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. unpow-1 73.7%

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

      2. times-frac 85.0%

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

   10. Simplified 85.0%

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

   if -2.1999999999999999e95 < y.re < -1.6e-139

   1. Initial program 83.5%

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

   if -1.6e-139 < y.re < 1.9999999999999999e-151

   1. Initial program 67.8%

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

      1. *-un-lft-identity 67.8%

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

      2. add-sqr-sqrt 67.8%

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

      3. times-frac 67.8%

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

      4. hypot-def 67.8%

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

      5. hypot-def 75.9%

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

   3. Applied egg-rr 75.9%

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

   4. Taylor expanded in y.im around -inf 49.7%

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

   5. Taylor expanded in y.im around -inf 97.3%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

   7. Applied egg-rr 97.2%

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

      1. unpow-1 97.2%

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

      2. associate-/r* 98.5%

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

   9. Simplified 98.5%

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

   if 1.9999999999999999e-151 < y.re < 4.79999999999999986e96

   1. Initial program 80.4%

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

      1. sub-neg 80.4%

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

      2. +-commutative 80.4%

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

      3. *-commutative 80.4%

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

      4. distribute-lft-neg-in 80.4%

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

      5. fma-def 80.4%

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

   3. Applied egg-rr 80.4%

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

   if 4.79999999999999986e96 < y.re

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.3%

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

      4. hypot-def 49.3%

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

      5. hypot-def 65.9%

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

   3. Applied egg-rr 65.9%

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

      1. associate-*l/ 66.2%

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

      2. *-un-lft-identity 66.2%

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

      3. *-commutative 66.2%

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

      4. *-commutative 66.2%

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

   5. Applied egg-rr 66.2%

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

   6. Taylor expanded in y.re around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. unsub-neg 81.4%

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

      3. *-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-/r* 81.5%

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

      6. div-sub 81.5%

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

      7. *-commutative 81.5%

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

      8. associate-*r/ 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.re y.im))))
   (if (<= y.im -8e+22)
     (/ (- x.re t_0) (hypot y.re y.im))
     (if (<= y.im 4e-138)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (if (<= y.im 1.6e+51)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (/ (- t_0 x.re) (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 / y_46_im);
	double tmp;
	if (y_46_im <= -8e+22) {
		tmp = (x_46_re - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 4e-138) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.6e+51) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (t_0 - x_46_re) / 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 / y_46_im);
	double tmp;
	if (y_46_im <= -8e+22) {
		tmp = (x_46_re - t_0) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 4e-138) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.6e+51) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (t_0 - x_46_re) / 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 / y_46_im)
	tmp = 0
	if y_46_im <= -8e+22:
		tmp = (x_46_re - t_0) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= 4e-138:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 1.6e+51:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (t_0 - x_46_re) / 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(x_46_im * Float64(y_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -8e+22)
		tmp = Float64(Float64(x_46_re - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 4e-138)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.6e+51)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(t_0 - x_46_re) / 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 / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -8e+22)
		tmp = (x_46_re - t_0) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= 4e-138)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 1.6e+51)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (t_0 - x_46_re) / 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[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8e+22], N[(N[(x$46$re - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4e-138], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+51], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8e22

   1. Initial program 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. add-sqr-sqrt 45.5%

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

      3. times-frac 45.5%

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

      4. hypot-def 45.5%

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

      5. hypot-def 59.2%

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

   3. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-un-lft-identity 59.3%

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

      3. *-commutative 59.3%

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

      4. *-commutative 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in y.im around -inf 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. associate-/l* 80.7%

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

      4. sub-neg 80.7%

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

      5. associate-/r/ 80.7%

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

   8. Simplified 80.7%

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

   if -8e22 < y.im < 4.00000000000000027e-138

   1. Initial program 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. add-sqr-sqrt 73.8%

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

      3. times-frac 73.8%

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

      4. hypot-def 73.8%

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

      5. hypot-def 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

   5. Applied egg-rr 85.6%

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

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

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
   7. 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. *-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. associate-/r* 83.2%

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

      6. div-sub 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*r/ 85.8%

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

   8. Simplified 85.8%

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

   if 4.00000000000000027e-138 < y.im < 1.6000000000000001e51

   1. Initial program 81.2%

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

   if 1.6000000000000001e51 < y.im

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.5%

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

      4. hypot-def 41.5%

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

      5. hypot-def 61.2%

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

   3. Applied egg-rr 61.2%

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

      1. associate-*l/ 61.3%

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

      2. *-un-lft-identity 61.3%

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

      3. *-commutative 61.3%

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

      4. *-commutative 61.3%

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

   5. Applied egg-rr 61.3%

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

   6. Taylor expanded in y.re around 0 76.8%

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

      1. associate-/l* 88.7%

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

      2. neg-mul-1 88.7%

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

      3. sub-neg 88.7%

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

      4. associate-/r/ 88.7%

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

   8. Simplified 88.7%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 10: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.35 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4e+22)
   (/ (- x.re (* x.im (/ y.re y.im))) (hypot y.re y.im))
   (if (<= y.im 7.2e-142)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (if (<= y.im 3.35e+50)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im 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 tmp;
	if (y_46_im <= -4e+22) {
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 7.2e-142) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.35e+50) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * 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 tmp;
	if (y_46_im <= -4e+22) {
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 7.2e-142) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.35e+50) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * 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):
	tmp = 0
	if y_46_im <= -4e+22:
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= 7.2e-142:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.35e+50:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * 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)
	tmp = 0.0
	if (y_46_im <= -4e+22)
		tmp = Float64(Float64(x_46_re - Float64(x_46_im * Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 7.2e-142)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.35e+50)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4e+22)
		tmp = (x_46_re - (x_46_im * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= 7.2e-142)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.35e+50)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * 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_] := If[LessEqual[y$46$im, -4e+22], N[(N[(x$46$re - N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.2e-142], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.35e+50], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4e22

   1. Initial program 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. add-sqr-sqrt 45.5%

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

      3. times-frac 45.5%

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

      4. hypot-def 45.5%

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

      5. hypot-def 59.2%

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

   3. Applied egg-rr 59.2%

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

      1. associate-*l/ 59.3%

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

      2. *-un-lft-identity 59.3%

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

      3. *-commutative 59.3%

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

      4. *-commutative 59.3%

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

   5. Applied egg-rr 59.3%

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

   6. Taylor expanded in y.im around -inf 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. associate-/l* 80.7%

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

      4. sub-neg 80.7%

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

      5. associate-/r/ 80.7%

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

   8. Simplified 80.7%

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

   if -4e22 < y.im < 7.20000000000000001e-142

   1. Initial program 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. add-sqr-sqrt 73.8%

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

      3. times-frac 73.8%

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

      4. hypot-def 73.8%

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

      5. hypot-def 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

   5. Applied egg-rr 85.6%

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

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

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
   7. 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. *-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. associate-/r* 83.2%

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

      6. div-sub 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*r/ 85.8%

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

   8. Simplified 85.8%

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

   if 7.20000000000000001e-142 < y.im < 3.3499999999999999e50

   1. Initial program 81.2%

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

   if 3.3499999999999999e50 < y.im

   1. Initial program 41.6%

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

   2. Taylor expanded in y.re around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. times-frac 88.1%

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

   4. Simplified 88.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re - x.im \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.35 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 11: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7e+74)
   (- (/ x.im y.re) (/ 1.0 (* (/ y.re y.im) (/ y.re x.re))))
   (if (<= y.re -1.9e+26)
     (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
     (if (<= y.re -2.2e-15)
       (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
       (if (<= y.re 3.5e-112)
         (* (/ -1.0 y.im) (+ x.re (/ -1.0 (/ (/ y.im y.re) x.im))))
         (/ (- x.im (* x.re (/ y.im y.re))) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7e+74) {
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -1.9e+26) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -2.2e-15) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_re <= 3.5e-112) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-7d+74)) then
        tmp = (x_46im / y_46re) - (1.0d0 / ((y_46re / y_46im) * (y_46re / x_46re)))
    else if (y_46re <= (-1.9d+26)) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= (-2.2d-15)) then
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) * (x_46re / y_46re))
    else if (y_46re <= 3.5d-112) then
        tmp = ((-1.0d0) / y_46im) * (x_46re + ((-1.0d0) / ((y_46im / y_46re) / x_46im)))
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7e+74) {
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	} else if (y_46_re <= -1.9e+26) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -2.2e-15) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_re <= 3.5e-112) {
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7e+74:
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)))
	elif y_46_re <= -1.9e+26:
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= -2.2e-15:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_re <= 3.5e-112:
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)))
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7e+74)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(1.0 / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / x_46_re))));
	elseif (y_46_re <= -1.9e+26)
		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 <= -2.2e-15)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	elseif (y_46_re <= 3.5e-112)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re + Float64(-1.0 / Float64(Float64(y_46_im / y_46_re) / x_46_im))));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7e+74)
		tmp = (x_46_im / y_46_re) - (1.0 / ((y_46_re / y_46_im) * (y_46_re / x_46_re)));
	elseif (y_46_re <= -1.9e+26)
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= -2.2e-15)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_re <= 3.5e-112)
		tmp = (-1.0 / y_46_im) * (x_46_re + (-1.0 / ((y_46_im / y_46_re) / x_46_im)));
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7e+74], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(1.0 / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.9e+26], 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, -2.2e-15], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.5e-112], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re + N[(-1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -7.00000000000000029e74

   1. Initial program 29.9%

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

   2. Taylor expanded in y.re around inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. unsub-neg 75.0%

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

      3. unpow2 75.0%

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

      4. times-frac 83.4%

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

   4. Simplified 83.4%

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

      1. frac-times 75.0%

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

      2. *-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. clear-num 74.9%

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

      2. inv-pow 74.9%

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

   8. Applied egg-rr 74.9%

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

      1. unpow-1 74.9%

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

      2. times-frac 83.4%

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

   10. Simplified 83.4%

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

   if -7.00000000000000029e74 < y.re < -1.9000000000000001e26

   1. Initial program 52.0%

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

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

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

      1. +-commutative 87.7%

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

      2. mul-1-neg 87.7%

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

      3. unsub-neg 87.7%

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

      4. unpow2 87.7%

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

      5. times-frac 87.9%

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

   4. Simplified 87.9%

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

   if -1.9000000000000001e26 < y.re < -2.19999999999999986e-15

   1. Initial program 99.1%

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

   2. Taylor expanded in y.re around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. unpow2 99.8%

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

      4. times-frac 100.0%

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

   4. Simplified 100.0%

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

   if -2.19999999999999986e-15 < y.re < 3.49999999999999994e-112

   1. Initial program 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. add-sqr-sqrt 75.0%

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

      3. times-frac 74.9%

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

      4. hypot-def 74.9%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y.im around -inf 45.0%

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

   5. Taylor expanded in y.im around -inf 88.8%

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

      1. clear-num 88.7%

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

      2. inv-pow 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. unpow-1 88.7%

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

      2. associate-/r* 89.6%

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

   9. Simplified 89.6%

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

   if 3.49999999999999994e-112 < y.re

   1. Initial program 62.4%

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

      1. *-un-lft-identity 62.4%

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

      2. add-sqr-sqrt 62.4%

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

      3. times-frac 62.3%

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

      4. hypot-def 62.3%

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

      5. hypot-def 77.0%

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

   3. Applied egg-rr 77.0%

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

      1. associate-*l/ 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. *-commutative 77.2%

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

      4. *-commutative 77.2%

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

   5. Applied egg-rr 77.2%

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

   6. Taylor expanded in y.re around inf 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unsub-neg 72.6%

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

      3. *-commutative 72.6%

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

      4. unpow2 72.6%

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

      5. associate-/r* 72.7%

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

      6. div-sub 72.6%

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

      7. *-commutative 72.6%

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

      8. associate-*r/ 73.8%

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

   8. Simplified 73.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{1}{\frac{y.re}{y.im} \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re + \frac{-1}{\frac{\frac{y.im}{y.re}}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 12: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -4e+22)
     t_0
     (if (<= y.im 3.5e-138)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (if (<= y.im 7.2e+50)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4e+22) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-138) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.2e+50) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-4d+22)) then
        tmp = t_0
    else if (y_46im <= 3.5d-138) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 7.2d+50) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4e+22) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-138) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 7.2e+50) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -4e+22:
		tmp = t_0
	elif y_46_im <= 3.5e-138:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 7.2e+50:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4e+22)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-138)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 7.2e+50)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4e+22)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-138)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 7.2e+50)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+22], t$95$0, If[LessEqual[y$46$im, 3.5e-138], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+50], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4e22 or 7.19999999999999972e50 < y.im

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around 0 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 84.9%

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

   4. Simplified 84.9%

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

   if -4e22 < y.im < 3.4999999999999999e-138

   1. Initial program 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. add-sqr-sqrt 73.8%

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

      3. times-frac 73.8%

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

      4. hypot-def 73.8%

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

      5. hypot-def 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. associate-*l/ 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

   5. Applied egg-rr 85.6%

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

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

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
   7. 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. *-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. associate-/r* 83.2%

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

      6. div-sub 85.1%

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

      7. *-commutative 85.1%

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

      8. associate-*r/ 85.8%

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

   8. Simplified 85.8%

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

   if 3.4999999999999999e-138 < y.im < 7.19999999999999972e50

   1. Initial program 81.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 13: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.95 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))))
   (if (<= y.im -2.95e+22)
     t_0
     (if (<= y.im 2e-70)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (if (<= y.im 2.7e-54)
         (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
         (if (<= y.im 1.1e+22)
           (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.95e+22) {
		tmp = t_0;
	} else if (y_46_im <= 2e-70) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.7e-54) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_im <= 1.1e+22) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    if (y_46im <= (-2.95d+22)) then
        tmp = t_0
    else if (y_46im <= 2d-70) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 2.7d-54) then
        tmp = ((-1.0d0) / y_46im) * (x_46re - ((y_46re * x_46im) / y_46im))
    else if (y_46im <= 1.1d+22) then
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) * (x_46re / y_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.95e+22) {
		tmp = t_0;
	} else if (y_46_im <= 2e-70) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.7e-54) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_im <= 1.1e+22) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.95e+22:
		tmp = t_0
	elif y_46_im <= 2e-70:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 2.7e-54:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_im <= 1.1e+22:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.95e+22)
		tmp = t_0;
	elseif (y_46_im <= 2e-70)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2.7e-54)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_im <= 1.1e+22)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.95e+22)
		tmp = t_0;
	elseif (y_46_im <= 2e-70)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 2.7e-54)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_im <= 1.1e+22)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.95e+22], t$95$0, If[LessEqual[y$46$im, 2e-70], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.7e-54], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+22], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.9500000000000001e22 or 1.1e22 < y.im

   1. Initial program 46.4%

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

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

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. unpow2 73.9%

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

      5. times-frac 83.9%

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

   4. Simplified 83.9%

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

   if -2.9500000000000001e22 < y.im < 1.99999999999999999e-70

   1. Initial program 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. add-sqr-sqrt 74.6%

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

      3. times-frac 74.5%

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

      4. hypot-def 74.5%

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

      5. hypot-def 86.6%

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

   3. Applied egg-rr 86.6%

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

      1. associate-*l/ 86.8%

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

      2. *-un-lft-identity 86.8%

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

      3. *-commutative 86.8%

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

      4. *-commutative 86.8%

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

   5. Applied egg-rr 86.8%

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

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

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
   7. 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. *-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. associate-/r* 82.6%

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

      6. div-sub 84.3%

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

      7. *-commutative 84.3%

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

      8. associate-*r/ 84.8%

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

   8. Simplified 84.8%

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

   if 1.99999999999999999e-70 < y.im < 2.70000000000000026e-54

   1. Initial program 87.1%

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

      1. *-un-lft-identity 87.1%

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

      2. add-sqr-sqrt 87.1%

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

      3. times-frac 86.9%

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

      4. hypot-def 86.9%

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

      5. hypot-def 87.3%

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

   3. Applied egg-rr 87.3%

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

   4. Taylor expanded in y.im around -inf 1.2%

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

   5. Taylor expanded in y.im around -inf 76.4%

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

   if 2.70000000000000026e-54 < y.im < 1.1e22

   1. Initial program 73.3%

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

   2. Taylor expanded in y.re around inf 61.1%

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

      1. mul-1-neg 61.1%

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

      2. unsub-neg 61.1%

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

      3. unpow2 61.1%

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

      4. times-frac 67.9%

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

   4. Simplified 67.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.95 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 14: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+22} \lor \neg \left(y.im \leq 1.45 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.6e+22) (not (<= y.im 1.45e+21)))
   (- (* (/ y.re y.im) (/ x.im y.im)) (/ x.re y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e+22) || !(y_46_im <= 1.45e+21)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.6d+22)) .or. (.not. (y_46im <= 1.45d+21))) then
        tmp = ((y_46re / y_46im) * (x_46im / y_46im)) - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e+22) || !(y_46_im <= 1.45e+21)) {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.6e+22) or not (y_46_im <= 1.45e+21):
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.6e+22) || !(y_46_im <= 1.45e+21))
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.6e+22) || ~((y_46_im <= 1.45e+21)))
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.6e+22], N[Not[LessEqual[y$46$im, 1.45e+21]], $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], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.6e22 or 1.45e21 < y.im

   1. Initial program 46.4%

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

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

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. unpow2 73.9%

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

      5. times-frac 83.9%

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

   4. Simplified 83.9%

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

   if -5.6e22 < y.im < 1.45e21

   1. Initial program 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. add-sqr-sqrt 75.1%

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

      3. times-frac 75.1%

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

      4. hypot-def 75.1%

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

      5. hypot-def 85.4%

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

   3. Applied egg-rr 85.4%

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

      1. associate-*l/ 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. *-commutative 85.6%

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

      4. *-commutative 85.6%

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

   5. Applied egg-rr 85.6%

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

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

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
   7. 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. *-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/r* 77.2%

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

      6. div-sub 78.7%

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

      7. *-commutative 78.7%

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

      8. associate-*r/ 79.8%

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

   8. Simplified 79.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+22} \lor \neg \left(y.im \leq 1.45 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 15: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+64} \lor \neg \left(y.im \leq 1.05 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.8e+64) (not (<= y.im 1.05e+22)))
   (/ (- x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.8e+64) || !(y_46_im <= 1.05e+22)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.8d+64)) .or. (.not. (y_46im <= 1.05d+22))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.8e+64) || !(y_46_im <= 1.05e+22)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.8e+64) or not (y_46_im <= 1.05e+22):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.8e+64) || !(y_46_im <= 1.05e+22))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.8e+64) || ~((y_46_im <= 1.05e+22)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.8e+64], N[Not[LessEqual[y$46$im, 1.05e+22]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.7999999999999996e64 or 1.0499999999999999e22 < y.im

   1. Initial program 45.2%

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

   2. Taylor expanded in y.re around 0 70.3%

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

      1. associate-*r/ 70.3%

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

      2. neg-mul-1 70.3%

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

   4. Simplified 70.3%

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

   if -7.7999999999999996e64 < y.im < 1.0499999999999999e22

   1. Initial program 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. add-sqr-sqrt 74.8%

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

      3. times-frac 74.7%

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

      4. hypot-def 74.7%

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

      5. hypot-def 84.7%

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

   3. Applied egg-rr 84.7%

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

      1. associate-*l/ 84.9%

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

      2. *-un-lft-identity 84.9%

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

      3. *-commutative 84.9%

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

      4. *-commutative 84.9%

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

   5. Applied egg-rr 84.9%

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

   6. Taylor expanded in y.re around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. unsub-neg 73.4%

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

      3. *-commutative 73.4%

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

      4. unpow2 73.4%

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

      5. associate-/r* 76.1%

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

      6. div-sub 77.5%

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

      7. *-commutative 77.5%

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

      8. associate-*r/ 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+64} \lor \neg \left(y.im \leq 1.05 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 16: 62.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.3e-21)
   (/ x.im y.re)
   (if (<= y.re 3.6e-112) (/ (- 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 <= -2.3e-21) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 3.6e-112) {
		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 <= (-2.3d-21)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 3.6d-112) 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 <= -2.3e-21) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 3.6e-112) {
		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 <= -2.3e-21:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 3.6e-112:
		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 <= -2.3e-21)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 3.6e-112)
		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 <= -2.3e-21)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 3.6e-112)
		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, -2.3e-21], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.6e-112], 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 < -2.29999999999999999e-21 or 3.6000000000000001e-112 < 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 65.0%

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

   if -2.29999999999999999e-21 < y.re < 3.6000000000000001e-112

   1. Initial program 75.0%

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

   2. Taylor expanded in y.re around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   4. Simplified 71.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 17: 44.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.4e+168) (/ x.re y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.4e+168) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5.4d+168)) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.4e+168) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.4e+168:
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.4e+168)
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.4e+168)
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.4e+168], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.40000000000000031e168

   1. Initial program 31.2%

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

      1. *-un-lft-identity 31.2%

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

      2. add-sqr-sqrt 31.2%

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

      3. times-frac 31.2%

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

      4. hypot-def 31.2%

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

      5. hypot-def 46.4%

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

   3. Applied egg-rr 46.4%

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

   4. Taylor expanded in y.im around -inf 86.0%

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

   5. Taylor expanded in y.re around 0 32.2%

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

   if -5.40000000000000031e168 < y.im

   1. Initial program 65.8%

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

   2. Taylor expanded in y.re around inf 46.6%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 18: 10.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

   1. *-un-lft-identity 63.1%

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

   2. add-sqr-sqrt 63.1%

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

   3. times-frac 63.1%

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

   4. hypot-def 63.1%

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

   5. hypot-def 75.8%

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

3. Applied egg-rr 75.8%

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

4. Taylor expanded in y.im around -inf 25.7%

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

5. Taylor expanded in y.re around -inf 8.1%

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

6. Final simplification 8.1%

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

Alternative 19: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

2. Taylor expanded in y.re around inf 43.7%

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

3. Final simplification 43.7%

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

Reproduce

herbie shell --seed 2023189 
(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))))