_divideComplex, imaginary part

Percentage Accurate: 62.0% → 91.9%

Time: 14.0s

Alternatives: 15

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \mathsf{fma}\left(t_0, t_1, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -2.85 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+183}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (hypot y.re y.im)))
        (t_1 (/ x.im (hypot y.re y.im)))
        (t_2 (fma t_0 t_1 (/ (- x.re) (/ (pow (hypot y.re y.im) 2.0) y.im)))))
   (if (<= y.im -2e+137)
     (fma t_0 t_1 (/ (- x.re) y.im))
     (if (<= y.im -2.85e-156)
       t_2
       (if (<= y.im 3.4e-167)
         (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
         (if (<= y.im 5.5e+183)
           t_2
           (/ (- (* y.re (/ x.im 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 t_0 = y_46_re / hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / hypot(y_46_re, y_46_im);
	double t_2 = fma(t_0, t_1, (-x_46_re / (pow(hypot(y_46_re, y_46_im), 2.0) / y_46_im)));
	double tmp;
	if (y_46_im <= -2e+137) {
		tmp = fma(t_0, t_1, (-x_46_re / y_46_im));
	} else if (y_46_im <= -2.85e-156) {
		tmp = t_2;
	} else if (y_46_im <= 3.4e-167) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if (y_46_im <= 5.5e+183) {
		tmp = t_2;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / 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_re / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	t_2 = fma(t_0, t_1, Float64(Float64(-x_46_re) / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2e+137)
		tmp = fma(t_0, t_1, Float64(Float64(-x_46_re) / y_46_im));
	elseif (y_46_im <= -2.85e-156)
		tmp = t_2;
	elseif (y_46_im <= 3.4e-167)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 5.5e+183)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / hypot(y_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[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + 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]}, If[LessEqual[y$46$im, -2e+137], N[(t$95$0 * t$95$1 + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.85e-156], t$95$2, If[LessEqual[y$46$im, 3.4e-167], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.5e+183], t$95$2, N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - 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 < -2.0000000000000001e137

   1. Initial program 31.0%

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

      1. div-sub 31.0%

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

      2. *-commutative 31.0%

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

      3. add-sqr-sqrt 31.0%

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

      4. times-frac 36.2%

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

      5. fma-neg 36.2%

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

      6. hypot-def 36.2%

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

      7. hypot-def 55.0%

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

      8. associate-/l* 58.8%

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

      9. add-sqr-sqrt 58.8%

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

      10. pow2 58.8%

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

      11. hypot-def 58.8%

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

   3. Applied egg-rr 58.8%

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

   4. Taylor expanded in y.re around 0 92.9%

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

   if -2.0000000000000001e137 < y.im < -2.84999999999999993e-156 or 3.3999999999999997e-167 < y.im < 5.5e183

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. *-commutative 66.8%

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

      3. add-sqr-sqrt 66.8%

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

      4. times-frac 71.8%

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

      5. fma-neg 71.8%

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

      6. hypot-def 71.8%

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

      7. hypot-def 88.6%

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

      8. associate-/l* 96.2%

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

      9. add-sqr-sqrt 96.2%

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

      10. pow2 96.2%

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

      11. hypot-def 96.2%

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

   3. Applied egg-rr 96.2%

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

   if -2.84999999999999993e-156 < y.im < 3.3999999999999997e-167

   1. Initial program 66.0%

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

      1. *-un-lft-identity 66.0%

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

      2. add-sqr-sqrt 66.0%

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

      3. times-frac 66.0%

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

      4. hypot-def 66.0%

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

      5. hypot-def 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y.re around inf 42.0%

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

   5. Taylor expanded in y.re around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. associate-*r/ 87.5%

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

   7. Simplified 87.5%

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

   if 5.5e183 < y.im

   1. Initial program 42.9%

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

      1. *-un-lft-identity 42.9%

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

      2. add-sqr-sqrt 42.9%

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

      3. times-frac 42.9%

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

      4. hypot-def 42.9%

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

      5. hypot-def 55.0%

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

   3. Applied egg-rr 55.0%

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

      1. associate-*l/ 55.1%

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

      2. *-un-lft-identity 55.1%

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

   5. Applied egg-rr 55.1%

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

   6. Taylor expanded in y.re around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. mul-1-neg 78.0%

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

      3. unsub-neg 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-*r/ 89.3%

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

   8. Simplified 89.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -2.85 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+183}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 89.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+302}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

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))) 1e+302)
     (/ (/ 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))) <= 1e+302) {
		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))) <= 1e+302)
		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], 1e+302], 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))) < 1.0000000000000001e302

   1. Initial program 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.3%

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

      4. hypot-def 79.3%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. associate-*l/ 95.4%

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

      2. *-un-lft-identity 95.4%

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

   5. Applied egg-rr 95.4%

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

   if 1.0000000000000001e302 < (/.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 6.8%

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

      1. div-sub 5.2%

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

      2. *-commutative 5.2%

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

      3. add-sqr-sqrt 5.2%

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

      4. times-frac 14.4%

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

      5. fma-neg 14.4%

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

      6. hypot-def 14.4%

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

      7. hypot-def 46.1%

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

      8. associate-/l* 60.5%

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

      9. add-sqr-sqrt 60.5%

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

      10. pow2 60.5%

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

      11. hypot-def 60.5%

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

   3. Applied egg-rr 60.5%

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

   4. Taylor expanded in y.re around 0 67.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+302}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternative 3: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq 10^{+302}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 1e+302)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (if (<= t_1 INFINITY)
       (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
       (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= 1e+302) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= 1e+302) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if t_1 <= 1e+302:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	elif t_1 <= math.inf:
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im)
	else:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_1 <= 1e+302)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (t_1 <= 1e+302)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	elseif (t_1 <= Inf)
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	else
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+302], 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], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.0000000000000001e302

   1. Initial program 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. add-sqr-sqrt 79.3%

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

      3. times-frac 79.3%

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

      4. hypot-def 79.3%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. associate-*l/ 95.4%

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

      2. *-un-lft-identity 95.4%

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

   5. Applied egg-rr 95.4%

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

   if 1.0000000000000001e302 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 22.5%

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

   2. Taylor expanded in y.re around 0 60.8%

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

      1. +-commutative 60.8%

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

      2. mul-1-neg 60.8%

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

      3. unsub-neg 60.8%

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

      4. *-commutative 60.8%

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

      5. unpow2 60.8%

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

      6. times-frac 70.2%

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

   4. Simplified 70.2%

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

      1. associate-*r/ 74.5%

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

   6. Applied egg-rr 74.5%

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

   if +inf.0 < (/.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 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. hypot-def 0.0%

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

      5. hypot-def 2.7%

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

   3. Applied egg-rr 2.7%

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

   4. Taylor expanded in y.re around inf 2.1%

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

   5. Taylor expanded in y.re around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. unsub-neg 45.7%

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

      3. associate-*r/ 59.3%

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

   7. Simplified 59.3%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+302}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := x.re \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{t_1 - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im - t_1\right) + -0.5 \cdot \left(\frac{x.im}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))) (t_1 (* x.re (/ y.im y.re))))
   (if (<= y.re -2.65e+121)
     (/ (- t_1 x.im) (hypot y.re y.im))
     (if (<= y.re -1.35e-25)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 8e-90)
         (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
         (if (<= y.re 2.3e+100)
           (/ t_0 (fma y.re y.re (* y.im y.im)))
           (/
            (+ (- x.im t_1) (* -0.5 (* (/ x.im y.re) (/ y.im (/ y.re y.im)))))
            (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.65e+121) {
		tmp = (t_1 - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.35e-25) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 8e-90) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.3e+100) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = ((x_46_im - t_1) + (-0.5 * ((x_46_im / y_46_re) * (y_46_im / (y_46_re / y_46_im))))) / 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(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(x_46_re * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.65e+121)
		tmp = Float64(Float64(t_1 - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.35e-25)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 8e-90)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.3e+100)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im - t_1) + Float64(-0.5 * Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / Float64(y_46_re / y_46_im))))) / hypot(y_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]}, Block[{t$95$1 = N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.65e+121], N[(N[(t$95$1 - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.35e-25], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8e-90], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+100], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im - t$95$1), $MachinePrecision] + N[(-0.5 * N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.65000000000000005e121

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 64.2%

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

   3. Applied egg-rr 64.2%

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

      1. associate-*l/ 64.3%

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

      2. *-un-lft-identity 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in y.re around -inf 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. associate-*r/ 95.2%

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

   8. Simplified 95.2%

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

   if -2.65000000000000005e121 < y.re < -1.35000000000000008e-25

   1. Initial program 80.0%

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

   if -1.35000000000000008e-25 < y.re < 7.99999999999999996e-90

   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. Taylor expanded in y.re around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. 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%

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

      1. associate-*r/ 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 7.99999999999999996e-90 < y.re < 2.2999999999999999e100

   1. Initial program 80.1%

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

      1. fma-def 80.2%

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

   3. Simplified 80.2%

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

   if 2.2999999999999999e100 < y.re

   1. Initial program 25.6%

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

      1. *-un-lft-identity 25.6%

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

      2. add-sqr-sqrt 25.6%

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

      3. times-frac 25.6%

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

      4. hypot-def 25.6%

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

      5. hypot-def 43.7%

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

   3. Applied egg-rr 43.7%

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

      1. associate-*l/ 43.7%

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

      2. *-un-lft-identity 43.7%

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

   5. Applied egg-rr 43.7%

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

   6. Taylor expanded in y.re around inf 57.9%

      \[\leadsto \frac{\color{blue}{x.im + \left(-1 \cdot \frac{x.re \cdot y.im}{y.re} + -0.5 \cdot \frac{x.im \cdot {y.im}^{2}}{{y.re}^{2}}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 57.9%

         \[\leadsto \frac{\color{blue}{\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right) + -0.5 \cdot \frac{x.im \cdot {y.im}^{2}}{{y.re}^{2}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. mul-1-neg 57.9%

         \[\leadsto \frac{\left(x.im + \color{blue}{\left(-\frac{x.re \cdot y.im}{y.re}\right)}\right) + -0.5 \cdot \frac{x.im \cdot {y.im}^{2}}{{y.re}^{2}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      3. unsub-neg 57.9%

         \[\leadsto \frac{\color{blue}{\left(x.im - \frac{x.re \cdot y.im}{y.re}\right)} + -0.5 \cdot \frac{x.im \cdot {y.im}^{2}}{{y.re}^{2}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      4. associate-*r/ 59.2%

         \[\leadsto \frac{\left(x.im - \color{blue}{x.re \cdot \frac{y.im}{y.re}}\right) + -0.5 \cdot \frac{x.im \cdot {y.im}^{2}}{{y.re}^{2}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      5. unpow2 59.2%

         \[\leadsto \frac{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \frac{x.im \cdot \color{blue}{\left(y.im \cdot y.im\right)}}{{y.re}^{2}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      6. unpow2 59.2%

         \[\leadsto \frac{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \frac{x.im \cdot \left(y.im \cdot y.im\right)}{\color{blue}{y.re \cdot y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      7. times-frac 68.3%

         \[\leadsto \frac{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \color{blue}{\left(\frac{x.im}{y.re} \cdot \frac{y.im \cdot y.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      8. associate-/l* 82.3%

         \[\leadsto \frac{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \left(\frac{x.im}{y.re} \cdot \color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   8. Simplified 82.3%

      \[\leadsto \frac{\color{blue}{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \left(\frac{x.im}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.re \cdot \frac{y.im}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right) + -0.5 \cdot \left(\frac{x.im}{y.re} \cdot \frac{y.im}{\frac{y.re}{y.im}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 82.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.re \cdot \frac{y.im}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= y.re -2.45e+121)
     (/ (- (* x.re (/ y.im y.re)) x.im) (hypot y.re y.im))
     (if (<= y.re -5e-22)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.8e-90)
         (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
         (if (<= y.re 1.65e+101)
           (/ t_0 (fma y.re y.re (* y.im y.im)))
           (- (/ x.im y.re) (/ y.im (/ y.re (/ x.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -2.45e+121) {
		tmp = ((x_46_re * (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -5e-22) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.8e-90) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.65e+101) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_re <= -2.45e+121)
		tmp = Float64(Float64(Float64(x_46_re * Float64(y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -5e-22)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.8e-90)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.65e+101)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re / Float64(x_46_re / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.45e+121], N[(N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5e-22], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-90], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+101], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re / N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.4499999999999999e121

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 64.2%

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

   3. Applied egg-rr 64.2%

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

      1. associate-*l/ 64.3%

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

      2. *-un-lft-identity 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in y.re around -inf 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. associate-*r/ 95.2%

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

   8. Simplified 95.2%

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

   if -2.4499999999999999e121 < y.re < -4.99999999999999954e-22

   1. Initial program 80.0%

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

   if -4.99999999999999954e-22 < y.re < 1.7999999999999999e-90

   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. Taylor expanded in y.re around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. 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%

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

      1. associate-*r/ 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 1.7999999999999999e-90 < y.re < 1.65000000000000006e101

   1. Initial program 80.1%

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

      1. fma-def 80.2%

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

   3. Simplified 80.2%

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

   if 1.65000000000000006e101 < y.re

   1. Initial program 25.6%

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

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

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/l* 74.7%

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

      6. associate-/r/ 74.7%

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

   4. Simplified 74.7%

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

      1. clear-num 74.7%

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

      2. inv-pow 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. unpow-1 74.7%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

      1. associate-*l/ 82.2%

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

      2. *-un-lft-identity 82.2%

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

   10. Applied egg-rr 82.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.re \cdot \frac{y.im}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-22}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-90}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 6: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.82 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.re \cdot \frac{y.im}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.82e+121)
     (/ (- (* x.re (/ y.im y.re)) x.im) (hypot y.re y.im))
     (if (<= y.re -6e-26)
       t_0
       (if (<= y.re 4.8e-89)
         (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
         (if (<= y.re 2.4e+100)
           t_0
           (- (/ x.im y.re) (/ y.im (/ y.re (/ x.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.82e+121) {
		tmp = ((x_46_re * (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -6e-26) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e-89) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.4e+100) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.82e+121) {
		tmp = ((x_46_re * (y_46_im / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -6e-26) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e-89) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 2.4e+100) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.82e+121:
		tmp = ((x_46_re * (y_46_im / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -6e-26:
		tmp = t_0
	elif y_46_re <= 4.8e-89:
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im)
	elif y_46_re <= 2.4e+100:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.82e+121)
		tmp = Float64(Float64(Float64(x_46_re * Float64(y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -6e-26)
		tmp = t_0;
	elseif (y_46_re <= 4.8e-89)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 2.4e+100)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re / Float64(x_46_re / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.82e+121)
		tmp = ((x_46_re * (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -6e-26)
		tmp = t_0;
	elseif (y_46_re <= 4.8e-89)
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	elseif (y_46_re <= 2.4e+100)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.82e+121], N[(N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6e-26], t$95$0, If[LessEqual[y$46$re, 4.8e-89], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+100], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re / N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.8199999999999999e121

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 64.2%

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

   3. Applied egg-rr 64.2%

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

      1. associate-*l/ 64.3%

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

      2. *-un-lft-identity 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in y.re around -inf 89.2%

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

      1. +-commutative 89.2%

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

      2. mul-1-neg 89.2%

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

      3. unsub-neg 89.2%

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

      4. associate-*r/ 95.2%

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

   8. Simplified 95.2%

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

   if -1.8199999999999999e121 < y.re < -6.00000000000000023e-26 or 4.80000000000000032e-89 < y.re < 2.40000000000000012e100

   1. Initial program 80.1%

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

   if -6.00000000000000023e-26 < y.re < 4.80000000000000032e-89

   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. Taylor expanded in y.re around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. 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%

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

      1. associate-*r/ 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 2.40000000000000012e100 < y.re

   1. Initial program 25.6%

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

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

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/l* 74.7%

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

      6. associate-/r/ 74.7%

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

   4. Simplified 74.7%

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

      1. clear-num 74.7%

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

      2. inv-pow 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. unpow-1 74.7%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

      1. associate-*l/ 82.2%

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

      2. *-un-lft-identity 82.2%

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

   10. Applied egg-rr 82.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.82 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.re \cdot \frac{y.im}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 7: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.8e+121)
     (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
     (if (<= y.re -5.2e-26)
       t_0
       (if (<= y.re 5.8e-92)
         (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
         (if (<= y.re 1e+100)
           t_0
           (- (/ x.im y.re) (/ y.im (/ y.re (/ x.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.8e+121) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if (y_46_re <= -5.2e-26) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-92) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1e+100) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.8d+121)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    else if (y_46re <= (-5.2d-26)) then
        tmp = t_0
    else if (y_46re <= 5.8d-92) then
        tmp = ((x_46im * (y_46re / y_46im)) / y_46im) - (x_46re / y_46im)
    else if (y_46re <= 1d+100) then
        tmp = t_0
    else
        tmp = (x_46im / y_46re) - (y_46im / (y_46re / (x_46re / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.8e+121) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if (y_46_re <= -5.2e-26) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-92) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1e+100) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.8e+121:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	elif y_46_re <= -5.2e-26:
		tmp = t_0
	elif y_46_re <= 5.8e-92:
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im)
	elif y_46_re <= 1e+100:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.8e+121)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= -5.2e-26)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-92)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1e+100)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re / Float64(x_46_re / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.8e+121)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	elseif (y_46_re <= -5.2e-26)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-92)
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	elseif (y_46_re <= 1e+100)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.8e+121], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-26], t$95$0, If[LessEqual[y$46$re, 5.8e-92], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e+100], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re / N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.80000000000000006e121

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 64.2%

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

   3. Applied egg-rr 64.2%

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

   4. Taylor expanded in y.re around inf 31.5%

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

   5. Taylor expanded in y.re around inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. unsub-neg 87.2%

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

      3. associate-*r/ 93.1%

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

   7. Simplified 93.1%

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

   if -2.80000000000000006e121 < y.re < -5.2000000000000002e-26 or 5.79999999999999969e-92 < y.re < 1.00000000000000002e100

   1. Initial program 80.1%

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

   if -5.2000000000000002e-26 < y.re < 5.79999999999999969e-92

   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. Taylor expanded in y.re around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. unpow2 80.5%

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

      6. 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%

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

      1. associate-*r/ 85.9%

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

   6. Applied egg-rr 85.9%

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

   if 1.00000000000000002e100 < y.re

   1. Initial program 25.6%

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

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

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/l* 74.7%

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

      6. associate-/r/ 74.7%

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

   4. Simplified 74.7%

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

      1. clear-num 74.7%

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

      2. inv-pow 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. unpow-1 74.7%

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

      2. associate-/l* 82.2%

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

   8. Simplified 82.2%

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

      1. associate-*l/ 82.2%

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

      2. *-un-lft-identity 82.2%

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

   10. Applied egg-rr 82.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 8: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.8 \cdot 10^{+54} \lor \neg \left(y.re \leq 10000000000 \lor \neg \left(y.re \leq 3 \cdot 10^{+19}\right) \land y.re \leq 9.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8.8e+54)
         (not
          (or (<= y.re 10000000000.0)
              (and (not (<= y.re 3e+19)) (<= y.re 9.5e+99)))))
   (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.8e+54) || !((y_46_re <= 10000000000.0) || (!(y_46_re <= 3e+19) && (y_46_re <= 9.5e+99)))) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-8.8d+54)) .or. (.not. (y_46re <= 10000000000.0d0) .or. (.not. (y_46re <= 3d+19)) .and. (y_46re <= 9.5d+99))) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.8e+54) || !((y_46_re <= 10000000000.0) || (!(y_46_re <= 3e+19) && (y_46_re <= 9.5e+99)))) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -8.8e+54) or not ((y_46_re <= 10000000000.0) or (not (y_46_re <= 3e+19) and (y_46_re <= 9.5e+99))):
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -8.8e+54) || !((y_46_re <= 10000000000.0) || (!(y_46_re <= 3e+19) && (y_46_re <= 9.5e+99))))
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -8.8e+54) || ~(((y_46_re <= 10000000000.0) || (~((y_46_re <= 3e+19)) && (y_46_re <= 9.5e+99)))))
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -8.8e+54], N[Not[Or[LessEqual[y$46$re, 10000000000.0], And[N[Not[LessEqual[y$46$re, 3e+19]], $MachinePrecision], LessEqual[y$46$re, 9.5e+99]]]], $MachinePrecision]], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.7999999999999996e54 or 1e10 < y.re < 3e19 or 9.49999999999999908e99 < y.re

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. add-sqr-sqrt 42.3%

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

      3. times-frac 42.3%

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

      4. hypot-def 42.3%

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

      5. hypot-def 59.1%

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

   3. Applied egg-rr 59.1%

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

   4. Taylor expanded in y.re around inf 34.7%

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

   5. Taylor expanded in y.re around inf 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

      3. associate-*r/ 84.9%

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

   7. Simplified 84.9%

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

   if -8.7999999999999996e54 < y.re < 1e10 or 3e19 < y.re < 9.49999999999999908e99

   1. Initial program 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. add-sqr-sqrt 71.3%

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

      3. times-frac 71.3%

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

      4. hypot-def 71.3%

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

      5. hypot-def 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. associate-*l/ 81.6%

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

      2. *-un-lft-identity 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in y.re around 0 73.3%

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

      1. neg-mul-1 73.3%

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

      2. +-commutative 73.3%

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

      3. *-commutative 73.3%

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

      4. unpow2 73.3%

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

      5. times-frac 80.3%

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

      6. fma-def 80.3%

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

      7. fma-neg 80.3%

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

      8. associate-*r/ 81.0%

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

      9. *-commutative 81.0%

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

      10. div-sub 82.4%

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

      11. /-rgt-identity 82.4%

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

      12. times-frac 80.4%

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

      13. *-commutative 80.4%

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

      14. times-frac 80.4%

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

      15. /-rgt-identity 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.8 \cdot 10^{+54} \lor \neg \left(y.re \leq 10000000000 \lor \neg \left(y.re \leq 3 \cdot 10^{+19}\right) \land y.re \leq 9.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 9: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 29000000000 \lor \neg \left(y.re \leq 1.05 \cdot 10^{+24}\right) \land y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.4e+52)
   (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
   (if (or (<= y.re 29000000000.0)
           (and (not (<= y.re 1.05e+24)) (<= y.re 1.35e+90)))
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (- (/ x.im y.re) (/ y.im (/ y.re (/ x.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 <= -4.4e+52) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if ((y_46_re <= 29000000000.0) || (!(y_46_re <= 1.05e+24) && (y_46_re <= 1.35e+90))) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_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 <= (-4.4d+52)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    else if ((y_46re <= 29000000000.0d0) .or. (.not. (y_46re <= 1.05d+24)) .and. (y_46re <= 1.35d+90)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im / y_46re) - (y_46im / (y_46re / (x_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 <= -4.4e+52) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if ((y_46_re <= 29000000000.0) || (!(y_46_re <= 1.05e+24) && (y_46_re <= 1.35e+90))) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_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 <= -4.4e+52:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	elif (y_46_re <= 29000000000.0) or (not (y_46_re <= 1.05e+24) and (y_46_re <= 1.35e+90)):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_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 <= -4.4e+52)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	elseif ((y_46_re <= 29000000000.0) || (!(y_46_re <= 1.05e+24) && (y_46_re <= 1.35e+90)))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re / Float64(x_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 <= -4.4e+52)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	elseif ((y_46_re <= 29000000000.0) || (~((y_46_re <= 1.05e+24)) && (y_46_re <= 1.35e+90)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_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, -4.4e+52], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 29000000000.0], And[N[Not[LessEqual[y$46$re, 1.05e+24]], $MachinePrecision], LessEqual[y$46$re, 1.35e+90]]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re / N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.4e52

   1. Initial program 49.9%

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

      1. *-un-lft-identity 49.9%

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

      2. add-sqr-sqrt 49.9%

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

      3. times-frac 49.9%

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

      4. hypot-def 49.9%

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

      5. hypot-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in y.re around inf 27.0%

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

   5. Taylor expanded in y.re around inf 83.6%

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

      1. mul-1-neg 83.6%

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

      2. unsub-neg 83.6%

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

      3. associate-*r/ 86.9%

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

   7. Simplified 86.9%

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

   if -4.4e52 < y.re < 2.9e10 or 1.0500000000000001e24 < y.re < 1.35e90

   1. Initial program 70.4%

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

      1. *-un-lft-identity 70.4%

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

      2. add-sqr-sqrt 70.4%

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

      3. times-frac 70.5%

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

      4. hypot-def 70.5%

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

      5. hypot-def 80.9%

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

   3. Applied egg-rr 80.9%

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

      1. associate-*l/ 81.0%

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

      2. *-un-lft-identity 81.0%

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

   5. Applied egg-rr 81.0%

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

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

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

      1. neg-mul-1 73.9%

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

      2. +-commutative 73.9%

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

      3. *-commutative 73.9%

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

      4. unpow2 73.9%

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

      5. times-frac 81.2%

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

      6. fma-def 81.2%

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

      7. fma-neg 81.2%

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

      8. associate-*r/ 81.9%

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

      9. *-commutative 81.9%

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

      10. div-sub 83.3%

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

      11. /-rgt-identity 83.3%

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

      12. times-frac 81.3%

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

      13. *-commutative 81.3%

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

      14. times-frac 81.2%

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

      15. /-rgt-identity 81.2%

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

   8. Simplified 81.2%

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

   if 2.9e10 < y.re < 1.0500000000000001e24 or 1.35e90 < y.re

   1. Initial program 37.9%

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

   2. Taylor expanded in y.re around inf 72.5%

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

      1. +-commutative 72.5%

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

      2. mul-1-neg 72.5%

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

      3. unsub-neg 72.5%

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

      4. unpow2 72.5%

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

      5. associate-/l* 73.2%

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

      6. associate-/r/ 75.0%

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

   4. Simplified 75.0%

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

      1. clear-num 75.0%

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

      2. inv-pow 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. unpow-1 75.0%

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

      2. associate-/l* 81.1%

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

   8. Simplified 81.1%

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

      1. associate-*l/ 81.2%

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

      2. *-un-lft-identity 81.2%

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

   10. Applied egg-rr 81.2%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 29000000000 \lor \neg \left(y.re \leq 1.05 \cdot 10^{+24}\right) \land y.re \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 10: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 38000000000:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+19} \lor \neg \left(y.re \leq 3.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.6e+53)
   (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))
   (if (<= y.re 38000000000.0)
     (- (/ (* x.im (/ y.re y.im)) y.im) (/ x.re y.im))
     (if (or (<= y.re 1.1e+19) (not (<= y.re 3.5e+92)))
       (- (/ x.im y.re) (/ y.im (/ y.re (/ x.re y.re))))
       (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.6e+53) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if (y_46_re <= 38000000000.0) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if ((y_46_re <= 1.1e+19) || !(y_46_re <= 3.5e+92)) {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.6d+53)) then
        tmp = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    else if (y_46re <= 38000000000.0d0) then
        tmp = ((x_46im * (y_46re / y_46im)) / y_46im) - (x_46re / y_46im)
    else if ((y_46re <= 1.1d+19) .or. (.not. (y_46re <= 3.5d+92))) then
        tmp = (x_46im / y_46re) - (y_46im / (y_46re / (x_46re / y_46re)))
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.6e+53) {
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	} else if (y_46_re <= 38000000000.0) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	} else if ((y_46_re <= 1.1e+19) || !(y_46_re <= 3.5e+92)) {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.6e+53:
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	elif y_46_re <= 38000000000.0:
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im)
	elif (y_46_re <= 1.1e+19) or not (y_46_re <= 3.5e+92):
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)))
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.6e+53)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 38000000000.0)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im) - Float64(x_46_re / y_46_im));
	elseif ((y_46_re <= 1.1e+19) || !(y_46_re <= 3.5e+92))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re / Float64(x_46_re / y_46_re))));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.6e+53)
		tmp = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	elseif (y_46_re <= 38000000000.0)
		tmp = ((x_46_im * (y_46_re / y_46_im)) / y_46_im) - (x_46_re / y_46_im);
	elseif ((y_46_re <= 1.1e+19) || ~((y_46_re <= 3.5e+92)))
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re / (x_46_re / y_46_re)));
	else
		tmp = ((y_46_re * (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$46$re, -1.6e+53], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 38000000000.0], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 1.1e+19], N[Not[LessEqual[y$46$re, 3.5e+92]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re / N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.6e53

   1. Initial program 49.9%

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

      1. *-un-lft-identity 49.9%

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

      2. add-sqr-sqrt 49.9%

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

      3. times-frac 49.9%

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

      4. hypot-def 49.9%

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

      5. hypot-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in y.re around inf 27.0%

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

   5. Taylor expanded in y.re around inf 83.6%

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

      1. mul-1-neg 83.6%

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

      2. unsub-neg 83.6%

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

      3. associate-*r/ 86.9%

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

   7. Simplified 86.9%

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

   if -1.6e53 < y.re < 3.8e10

   1. Initial program 72.1%

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

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

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. *-commutative 76.6%

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

      5. unpow2 76.6%

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

      6. times-frac 81.6%

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

   4. Simplified 81.6%

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

      1. associate-*r/ 82.3%

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

   6. Applied egg-rr 82.3%

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

   if 3.8e10 < y.re < 1.1e19 or 3.49999999999999986e92 < y.re

   1. Initial program 37.9%

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

   2. Taylor expanded in y.re around inf 72.5%

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

      1. +-commutative 72.5%

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

      2. mul-1-neg 72.5%

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

      3. unsub-neg 72.5%

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

      4. unpow2 72.5%

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

      5. associate-/l* 73.2%

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

      6. associate-/r/ 75.0%

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

   4. Simplified 75.0%

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

      1. clear-num 75.0%

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

      2. inv-pow 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. unpow-1 75.0%

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

      2. associate-/l* 81.1%

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

   8. Simplified 81.1%

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

      1. associate-*l/ 81.2%

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

      2. *-un-lft-identity 81.2%

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

   10. Applied egg-rr 81.2%

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

   if 1.1e19 < y.re < 3.49999999999999986e92

   1. Initial program 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. add-sqr-sqrt 52.5%

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

      3. times-frac 52.6%

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

      4. hypot-def 52.6%

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

      5. hypot-def 67.8%

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

   3. Applied egg-rr 67.8%

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

      1. associate-*l/ 67.9%

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

      2. *-un-lft-identity 67.9%

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

   5. Applied egg-rr 67.9%

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

   6. Taylor expanded in y.re around 0 45.9%

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

      1. neg-mul-1 45.9%

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

      2. +-commutative 45.9%

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

      3. *-commutative 45.9%

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

      4. unpow2 45.9%

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

      5. times-frac 77.0%

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

      6. fma-def 77.0%

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

      7. fma-neg 77.0%

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

      8. associate-*r/ 77.2%

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

      9. *-commutative 77.2%

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

      10. div-sub 77.2%

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

      11. /-rgt-identity 77.2%

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

      12. times-frac 61.1%

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

      13. *-commutative 61.1%

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

      14. times-frac 77.2%

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

      15. /-rgt-identity 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 38000000000:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+19} \lor \neg \left(y.re \leq 3.5 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{y.re}{\frac{x.re}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 11: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ t_1 := \frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 12200000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* y.re (/ x.im y.im)) x.re) y.im))
        (t_1 (* (/ 1.0 y.re) (- x.im (* x.re (/ y.im y.re))))))
   (if (<= y.re -6.2e+52)
     t_1
     (if (<= y.re 12200000000.0)
       t_0
       (if (<= y.re 5.2e+19)
         (- (/ x.im y.re) (* y.im (/ x.re (* y.re y.re))))
         (if (<= y.re 1.2e+100) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -6.2e+52) {
		tmp = t_1;
	} else if (y_46_re <= 12200000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 5.2e+19) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= 1.2e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    t_1 = (1.0d0 / y_46re) * (x_46im - (x_46re * (y_46im / y_46re)))
    if (y_46re <= (-6.2d+52)) then
        tmp = t_1
    else if (y_46re <= 12200000000.0d0) then
        tmp = t_0
    else if (y_46re <= 5.2d+19) then
        tmp = (x_46im / y_46re) - (y_46im * (x_46re / (y_46re * y_46re)))
    else if (y_46re <= 1.2d+100) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -6.2e+52) {
		tmp = t_1;
	} else if (y_46_re <= 12200000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 5.2e+19) {
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	} else if (y_46_re <= 1.2e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	t_1 = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -6.2e+52:
		tmp = t_1
	elif y_46_re <= 12200000000.0:
		tmp = t_0
	elif y_46_re <= 5.2e+19:
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)))
	elif y_46_re <= 1.2e+100:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	t_1 = Float64(Float64(1.0 / y_46_re) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -6.2e+52)
		tmp = t_1;
	elseif (y_46_re <= 12200000000.0)
		tmp = t_0;
	elseif (y_46_re <= 5.2e+19)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(x_46_re / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 1.2e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	t_1 = (1.0 / y_46_re) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -6.2e+52)
		tmp = t_1;
	elseif (y_46_re <= 12200000000.0)
		tmp = t_0;
	elseif (y_46_re <= 5.2e+19)
		tmp = (x_46_im / y_46_re) - (y_46_im * (x_46_re / (y_46_re * y_46_re)));
	elseif (y_46_re <= 1.2e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e+52], t$95$1, If[LessEqual[y$46$re, 12200000000.0], t$95$0, If[LessEqual[y$46$re, 5.2e+19], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.2e+100], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.2e52 or 1.20000000000000006e100 < y.re

   1. Initial program 39.7%

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

      1. *-un-lft-identity 39.7%

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

      2. add-sqr-sqrt 39.7%

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

      3. times-frac 39.7%

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

      4. hypot-def 39.7%

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

      5. hypot-def 57.2%

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

   3. Applied egg-rr 57.2%

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

   4. Taylor expanded in y.re around inf 31.9%

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

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

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

      1. mul-1-neg 79.2%

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

      2. unsub-neg 79.2%

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

      3. associate-*r/ 84.5%

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

   7. Simplified 84.5%

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

   if -6.2e52 < y.re < 1.22e10 or 5.2e19 < y.re < 1.20000000000000006e100

   1. Initial program 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. add-sqr-sqrt 71.3%

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

      3. times-frac 71.3%

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

      4. hypot-def 71.3%

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

      5. hypot-def 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. associate-*l/ 81.6%

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

      2. *-un-lft-identity 81.6%

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

   5. Applied egg-rr 81.6%

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

   6. Taylor expanded in y.re around 0 73.3%

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

      1. neg-mul-1 73.3%

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

      2. +-commutative 73.3%

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

      3. *-commutative 73.3%

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

      4. unpow2 73.3%

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

      5. times-frac 80.3%

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

      6. fma-def 80.3%

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

      7. fma-neg 80.3%

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

      8. associate-*r/ 81.0%

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

      9. *-commutative 81.0%

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

      10. div-sub 82.4%

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

      11. /-rgt-identity 82.4%

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

      12. times-frac 80.4%

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

      13. *-commutative 80.4%

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

      14. times-frac 80.4%

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

      15. /-rgt-identity 80.4%

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

   8. Simplified 80.4%

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

   if 1.22e10 < y.re < 5.2e19

   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 95.2%

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

      1. +-commutative 95.2%

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

      2. mul-1-neg 95.2%

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

      3. unsub-neg 95.2%

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

      4. unpow2 95.2%

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

      5. associate-/l* 94.9%

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

      6. associate-/r/ 95.2%

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

   4. Simplified 95.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 12200000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 12: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+76} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8.5e+76) (not (<= y.re 1.15e+100)))
   (/ x.im y.re)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.5e+76) || !(y_46_re <= 1.15e+100)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-8.5d+76)) .or. (.not. (y_46re <= 1.15d+100))) then
        tmp = x_46im / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -8.5e+76) || !(y_46_re <= 1.15e+100)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -8.5e+76) or not (y_46_re <= 1.15e+100):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -8.5e+76) || !(y_46_re <= 1.15e+100))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -8.5e+76) || ~((y_46_re <= 1.15e+100)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -8.5e+76], N[Not[LessEqual[y$46$re, 1.15e+100]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.49999999999999992e76 or 1.14999999999999995e100 < y.re

   1. Initial program 38.6%

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

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

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

   if -8.49999999999999992e76 < y.re < 1.14999999999999995e100

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.9%

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

      4. hypot-def 71.9%

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

      5. hypot-def 81.4%

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

   3. Applied egg-rr 81.4%

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

      1. associate-*l/ 81.5%

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

      2. *-un-lft-identity 81.5%

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

   5. Applied egg-rr 81.5%

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

   6. Taylor expanded in y.re around 0 70.0%

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

      1. neg-mul-1 70.0%

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

      2. +-commutative 70.0%

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

      3. *-commutative 70.0%

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

      4. unpow2 70.0%

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

      5. times-frac 76.7%

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

      6. fma-def 76.7%

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

      7. fma-neg 76.7%

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

      8. associate-*r/ 77.3%

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

      9. *-commutative 77.3%

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

      10. div-sub 78.6%

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

      11. /-rgt-identity 78.6%

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

      12. times-frac 76.8%

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

      13. *-commutative 76.8%

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

      14. times-frac 76.7%

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

      15. /-rgt-identity 76.7%

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

   8. Simplified 76.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+76} \lor \neg \left(y.re \leq 1.15 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 13: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.55e+19)
   (/ x.im y.re)
   (if (<= y.re 1.15e+99) (/ (- 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 <= -1.55e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.15e+99) {
		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 <= (-1.55d+19)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 1.15d+99) 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 <= -1.55e+19) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.15e+99) {
		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 <= -1.55e+19:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 1.15e+99:
		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 <= -1.55e+19)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.15e+99)
		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 <= -1.55e+19)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 1.15e+99)
		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, -1.55e+19], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+99], 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 < -1.55e19 or 1.1500000000000001e99 < y.re

   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 inf 65.6%

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

   if -1.55e19 < y.re < 1.1500000000000001e99

   1. Initial program 71.5%

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

   2. Taylor expanded in y.re around 0 66.0%

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

      1. associate-*r/ 66.0%

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

      2. neg-mul-1 66.0%

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

   4. Simplified 66.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 14: 9.2% 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 58.6%

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

   1. *-un-lft-identity 58.6%

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

   2. add-sqr-sqrt 58.6%

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

   3. times-frac 58.6%

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

   4. hypot-def 58.6%

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

   5. hypot-def 71.6%

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

3. Applied egg-rr 71.6%

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

4. Taylor expanded in y.re around -inf 27.5%

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

   1. mul-1-neg 27.5%

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

6. Simplified 27.5%

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

7. Taylor expanded in y.im around -inf 11.3%

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

8. Final simplification 11.3%

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

Alternative 15: 42.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in y.re around inf 39.7%

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

3. Final simplification 39.7%

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

Reproduce

herbie shell --seed 2023293 
(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))))