_divideComplex, imaginary part

Percentage Accurate: 61.2% → 92.0%

Time: 14.3s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% 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: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{x.im}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+159}:\\ \;\;\;\;t_1 \cdot \left(x.re - t_2\right)\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ (- x.re) (/ (pow (hypot y.re y.im) 2.0) y.im))))
        (t_1 (/ 1.0 (hypot y.re y.im)))
        (t_2 (/ x.im (/ y.im y.re))))
   (if (<= y.im -5e+159)
     (* t_1 (- x.re t_2))
     (if (<= y.im -5.6e-189)
       t_0
       (if (<= y.im 4.5e-161)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 1.35e+154) t_0 (* t_1 (- t_2 x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / (pow(hypot(y_46_re, y_46_im), 2.0) / y_46_im)));
	double t_1 = 1.0 / hypot(y_46_re, y_46_im);
	double t_2 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -5e+159) {
		tmp = t_1 * (x_46_re - t_2);
	} else if (y_46_im <= -5.6e-189) {
		tmp = t_0;
	} else if (y_46_im <= 4.5e-161) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1 * (t_2 - x_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(-x_46_re) / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_im)))
	t_1 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_2 = Float64(x_46_im / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -5e+159)
		tmp = Float64(t_1 * Float64(x_46_re - t_2));
	elseif (y_46_im <= -5.6e-189)
		tmp = t_0;
	elseif (y_46_im <= 4.5e-161)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(t_1 * Float64(t_2 - x_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 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-x$46$re) / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5e+159], N[(t$95$1 * N[(x$46$re - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.6e-189], t$95$0, If[LessEqual[y$46$im, 4.5e-161], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+154], t$95$0, N[(t$95$1 * N[(t$95$2 - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.00000000000000003e159

   1. Initial program 24.0%

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

      1. *-un-lft-identity 24.0%

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

      2. add-sqr-sqrt 24.0%

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

      3. times-frac 24.0%

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

      4. hypot-def 24.0%

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

      5. hypot-def 63.4%

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

   4. Applied egg-rr 63.4%

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

   5. Taylor expanded in y.im around -inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. associate-/l* 92.5%

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

   7. Simplified 92.5%

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

   if -5.00000000000000003e159 < y.im < -5.5999999999999999e-189 or 4.4999999999999996e-161 < y.im < 1.35000000000000003e154

   1. Initial program 78.5%

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

      1. div-sub 78.5%

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

      2. sub-neg 78.5%

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

      3. *-commutative 78.5%

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

      4. add-sqr-sqrt 78.5%

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

      5. times-frac 79.7%

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

      6. fma-def 79.7%

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

      7. hypot-def 79.7%

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

      8. hypot-def 90.9%

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

      9. associate-/l* 94.1%

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

      10. add-sqr-sqrt 94.1%

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

      11. pow2 94.1%

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

      12. hypot-def 94.1%

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

   4. Applied egg-rr 94.1%

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

   if -5.5999999999999999e-189 < y.im < 4.4999999999999996e-161

   1. Initial program 73.9%

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

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

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

      4. associate-/l* 85.4%

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

      5. associate-/r/ 83.8%

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

   5. Simplified 83.8%

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

      1. pow2 83.8%

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

      2. associate-*l/ 83.6%

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

      3. associate-/r* 94.6%

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

   7. Applied egg-rr 94.6%

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

      1. associate-/l/ 83.6%

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

      2. times-frac 91.9%

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

   9. Applied egg-rr 91.9%

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

       1. associate-*r/ 91.9%

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

       2. associate-/r/ 95.1%

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

   11. Simplified 95.1%

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

   if 1.35000000000000003e154 < y.im

   1. Initial program 35.0%

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

      1. *-un-lft-identity 35.0%

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

      2. add-sqr-sqrt 35.0%

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

      3. times-frac 35.0%

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

      4. hypot-def 35.0%

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

      5. hypot-def 56.7%

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

   4. Applied egg-rr 56.7%

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

   5. Taylor expanded in y.re around 0 92.5%

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

      1. neg-mul-1 92.5%

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

      2. +-commutative 92.5%

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

      3. unsub-neg 92.5%

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

      4. associate-/l* 95.4%

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

   7. Simplified 95.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ t_2 := \frac{x.im}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+149}:\\ \;\;\;\;t_0 \cdot \left(x.re - t_2\right)\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_2 - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1
         (-
          (* t_0 (/ y.re (/ (hypot y.re y.im) x.im)))
          (* y.im (/ x.re (pow (hypot y.re y.im) 2.0)))))
        (t_2 (/ x.im (/ y.im y.re))))
   (if (<= y.im -3.6e+149)
     (* t_0 (- x.re t_2))
     (if (<= y.im -1.9e-86)
       t_1
       (if (<= y.im 1.9e-89)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 2.35e+147) t_1 (* t_0 (- t_2 x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / pow(hypot(y_46_re, y_46_im), 2.0)));
	double t_2 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.6e+149) {
		tmp = t_0 * (x_46_re - t_2);
	} else if (y_46_im <= -1.9e-86) {
		tmp = t_1;
	} else if (y_46_im <= 1.9e-89) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 2.35e+147) {
		tmp = t_1;
	} else {
		tmp = t_0 * (t_2 - x_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 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (t_0 * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0)));
	double t_2 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.6e+149) {
		tmp = t_0 * (x_46_re - t_2);
	} else if (y_46_im <= -1.9e-86) {
		tmp = t_1;
	} else if (y_46_im <= 1.9e-89) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 2.35e+147) {
		tmp = t_1;
	} else {
		tmp = t_0 * (t_2 - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (t_0 * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / math.pow(math.hypot(y_46_re, y_46_im), 2.0)))
	t_2 = x_46_im / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -3.6e+149:
		tmp = t_0 * (x_46_re - t_2)
	elif y_46_im <= -1.9e-86:
		tmp = t_1
	elif y_46_im <= 1.9e-89:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_im <= 2.35e+147:
		tmp = t_1
	else:
		tmp = t_0 * (t_2 - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(t_0 * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(y_46_im * Float64(x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0))))
	t_2 = Float64(x_46_im / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.6e+149)
		tmp = Float64(t_0 * Float64(x_46_re - t_2));
	elseif (y_46_im <= -1.9e-86)
		tmp = t_1;
	elseif (y_46_im <= 1.9e-89)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 2.35e+147)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(t_2 - x_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 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (t_0 * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im))) - (y_46_im * (x_46_re / (hypot(y_46_re, y_46_im) ^ 2.0)));
	t_2 = x_46_im / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -3.6e+149)
		tmp = t_0 * (x_46_re - t_2);
	elseif (y_46_im <= -1.9e-86)
		tmp = t_1;
	elseif (y_46_im <= 1.9e-89)
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_im <= 2.35e+147)
		tmp = t_1;
	else
		tmp = t_0 * (t_2 - x_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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[(x$46$re / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+149], N[(t$95$0 * N[(x$46$re - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.9e-86], t$95$1, If[LessEqual[y$46$im, 1.9e-89], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.35e+147], t$95$1, N[(t$95$0 * N[(t$95$2 - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.59999999999999995e149

   1. Initial program 27.0%

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

      1. *-un-lft-identity 27.0%

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

      2. add-sqr-sqrt 27.0%

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

      3. times-frac 27.1%

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

      4. hypot-def 27.1%

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

      5. hypot-def 64.7%

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

   4. Applied egg-rr 64.7%

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

   5. Taylor expanded in y.im around -inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. unsub-neg 89.3%

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

      3. associate-/l* 90.6%

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

   7. Simplified 90.6%

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

   if -3.59999999999999995e149 < y.im < -1.9e-86 or 1.9000000000000001e-89 < y.im < 2.3500000000000001e147

   1. Initial program 80.8%

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

      1. div-sub 80.7%

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

      2. sub-neg 80.7%

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

      3. *-un-lft-identity 80.7%

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

      4. add-sqr-sqrt 80.7%

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

      5. times-frac 80.7%

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

      6. fma-def 80.7%

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

      7. hypot-def 80.7%

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

      8. hypot-def 82.0%

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

      9. associate-/l* 86.1%

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

      10. add-sqr-sqrt 86.1%

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

      11. pow2 86.1%

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

      12. hypot-def 86.1%

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

   4. Applied egg-rr 86.1%

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

      1. fma-neg 86.1%

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

      2. *-commutative 86.1%

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

      3. associate-/l* 96.0%

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

      4. associate-/r/ 93.7%

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

      5. *-commutative 93.7%

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

   6. Simplified 93.7%

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

   if -1.9e-86 < y.im < 1.9000000000000001e-89

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around inf 84.0%

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

      1. +-commutative 84.0%

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

      2. mul-1-neg 84.0%

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

      3. unsub-neg 84.0%

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

      4. associate-/l* 85.3%

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

      5. associate-/r/ 83.1%

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

   5. Simplified 83.1%

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

      1. pow2 83.1%

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

      2. associate-*l/ 84.0%

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

      3. associate-/r* 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. associate-/l/ 84.0%

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

      2. times-frac 88.8%

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

   9. Applied egg-rr 88.8%

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

       1. associate-*r/ 88.7%

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

       2. associate-/r/ 92.0%

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

   11. Simplified 92.0%

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

   if 2.3500000000000001e147 < y.im

   1. Initial program 36.7%

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

      1. *-un-lft-identity 36.7%

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

      2. add-sqr-sqrt 36.7%

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

      3. times-frac 36.7%

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

      4. hypot-def 36.7%

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

      5. hypot-def 57.9%

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

   4. Applied egg-rr 57.9%

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

   5. Taylor expanded in y.re around 0 92.6%

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

      1. neg-mul-1 92.6%

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

      2. +-commutative 92.6%

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

      3. unsub-neg 92.6%

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

      4. associate-/l* 95.5%

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

   7. Simplified 95.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - y.im \cdot \frac{x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

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))) 2e+202)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (* (- (/ x.im (/ y.im y.re)) x.re) (/ 1.0 y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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))) <= 2e+202) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+202) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+202:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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))) <= 2e+202)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) * Float64(1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+202)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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], 2e+202], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / 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))) < 1.9999999999999998e202

   1. Initial program 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. add-sqr-sqrt 80.4%

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

      3. times-frac 80.4%

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

      4. hypot-def 80.4%

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

      5. hypot-def 96.7%

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

   4. Applied egg-rr 96.7%

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

   if 1.9999999999999998e202 < (/.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 13.7%

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

      1. *-un-lft-identity 13.7%

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

      2. add-sqr-sqrt 13.7%

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

      3. times-frac 13.7%

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

      4. hypot-def 13.7%

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

      5. hypot-def 18.4%

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

   4. Applied egg-rr 18.4%

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

   5. Taylor expanded in y.re around 0 34.0%

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

      1. neg-mul-1 34.0%

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

      2. +-commutative 34.0%

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

      3. unsub-neg 34.0%

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

      4. associate-/l* 35.8%

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

   7. Simplified 35.8%

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

   8. Taylor expanded in y.re around 0 60.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{-1} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -4.7e+83)
     (* (/ (/ y.im (hypot y.re y.im)) -1.0) (/ x.re (hypot y.re y.im)))
     (if (<= y.im -3.8e-105)
       t_0
       (if (<= y.im 5.2e-114)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 1.1e+97)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (- (/ x.im (/ y.im y.re)) x.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_im <= -4.7e+83) {
		tmp = ((y_46_im / hypot(y_46_re, y_46_im)) / -1.0) * (x_46_re / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -3.8e-105) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 1.1e+97) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_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_im <= -4.7e+83) {
		tmp = ((y_46_im / Math.hypot(y_46_re, y_46_im)) / -1.0) * (x_46_re / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -3.8e-105) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 1.1e+97) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_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_im <= -4.7e+83:
		tmp = ((y_46_im / math.hypot(y_46_re, y_46_im)) / -1.0) * (x_46_re / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -3.8e-105:
		tmp = t_0
	elif y_46_im <= 5.2e-114:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_im <= 1.1e+97:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_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_im <= -4.7e+83)
		tmp = Float64(Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) / -1.0) * Float64(x_46_re / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -3.8e-105)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-114)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 1.1e+97)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_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_im <= -4.7e+83)
		tmp = ((y_46_im / hypot(y_46_re, y_46_im)) / -1.0) * (x_46_re / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -3.8e-105)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-114)
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_im <= 1.1e+97)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_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$46$im, -4.7e+83], N[(N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-105], t$95$0, If[LessEqual[y$46$im, 5.2e-114], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+97], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.6999999999999999e83

   1. Initial program 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. add-sqr-sqrt 38.2%

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

      3. times-frac 38.2%

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

      4. hypot-def 38.2%

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

      5. hypot-def 66.1%

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

   4. Applied egg-rr 66.1%

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

   5. Taylor expanded in x.im around 0 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

      3. *-commutative 63.1%

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

   7. Simplified 63.1%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

      3. frac-2neg 63.3%

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

   9. Applied egg-rr 84.5%

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

       1. associate-/r/ 85.7%

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

       2. neg-mul-1 85.7%

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

       3. times-frac 85.7%

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

   11. Applied egg-rr 85.7%

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

   if -4.6999999999999999e83 < y.im < -3.7999999999999998e-105 or 5.20000000000000026e-114 < y.im < 1.1e97

   1. Initial program 85.9%

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

   if -3.7999999999999998e-105 < y.im < 5.20000000000000026e-114

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. associate-/l* 87.7%

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

      5. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

      1. pow2 85.4%

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

      2. associate-*l/ 86.3%

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

      3. associate-/r* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. associate-/l/ 86.3%

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

      2. times-frac 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. associate-*r/ 91.5%

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

       2. associate-/r/ 95.1%

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

   11. Simplified 95.1%

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

   if 1.1e97 < y.im

   1. Initial program 47.1%

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

      1. *-un-lft-identity 47.1%

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

      2. add-sqr-sqrt 47.1%

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

      3. times-frac 47.1%

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

      4. hypot-def 47.1%

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

      5. hypot-def 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in y.re around 0 92.2%

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

      1. neg-mul-1 92.2%

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

      2. +-commutative 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 92.6%

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

   7. Simplified 92.6%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{-1} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{x.im}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+84}:\\ \;\;\;\;t_1 \cdot \left(x.re - t_2\right)\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_2 - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ 1.0 (hypot y.re y.im)))
        (t_2 (/ x.im (/ y.im y.re))))
   (if (<= y.im -7e+84)
     (* t_1 (- x.re t_2))
     (if (<= y.im -4e-102)
       t_0
       (if (<= y.im 1.45e-113)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 1.7e+93) t_0 (* t_1 (- t_2 x.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 t_1 = 1.0 / hypot(y_46_re, y_46_im);
	double t_2 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -7e+84) {
		tmp = t_1 * (x_46_re - t_2);
	} else if (y_46_im <= -4e-102) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e-113) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 1.7e+93) {
		tmp = t_0;
	} else {
		tmp = t_1 * (t_2 - x_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 t_1 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_2 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -7e+84) {
		tmp = t_1 * (x_46_re - t_2);
	} else if (y_46_im <= -4e-102) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e-113) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 1.7e+93) {
		tmp = t_0;
	} else {
		tmp = t_1 * (t_2 - x_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))
	t_1 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_2 = x_46_im / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -7e+84:
		tmp = t_1 * (x_46_re - t_2)
	elif y_46_im <= -4e-102:
		tmp = t_0
	elif y_46_im <= 1.45e-113:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_im <= 1.7e+93:
		tmp = t_0
	else:
		tmp = t_1 * (t_2 - x_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)))
	t_1 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_2 = Float64(x_46_im / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -7e+84)
		tmp = Float64(t_1 * Float64(x_46_re - t_2));
	elseif (y_46_im <= -4e-102)
		tmp = t_0;
	elseif (y_46_im <= 1.45e-113)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 1.7e+93)
		tmp = t_0;
	else
		tmp = Float64(t_1 * Float64(t_2 - x_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));
	t_1 = 1.0 / hypot(y_46_re, y_46_im);
	t_2 = x_46_im / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -7e+84)
		tmp = t_1 * (x_46_re - t_2);
	elseif (y_46_im <= -4e-102)
		tmp = t_0;
	elseif (y_46_im <= 1.45e-113)
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_im <= 1.7e+93)
		tmp = t_0;
	else
		tmp = t_1 * (t_2 - x_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]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7e+84], N[(t$95$1 * N[(x$46$re - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4e-102], t$95$0, If[LessEqual[y$46$im, 1.45e-113], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+93], t$95$0, N[(t$95$1 * N[(t$95$2 - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.9999999999999998e84

   1. Initial program 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. add-sqr-sqrt 37.0%

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

      3. times-frac 37.0%

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

      4. hypot-def 37.0%

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

      5. hypot-def 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in y.im around -inf 83.1%

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

      1. mul-1-neg 83.1%

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

      2. unsub-neg 83.1%

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

      3. associate-/l* 84.3%

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

   7. Simplified 84.3%

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

   if -6.9999999999999998e84 < y.im < -3.99999999999999973e-102 or 1.45000000000000002e-113 < y.im < 1.7e93

   1. Initial program 86.0%

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

   if -3.99999999999999973e-102 < y.im < 1.45000000000000002e-113

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. associate-/l* 87.7%

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

      5. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

      1. pow2 85.4%

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

      2. associate-*l/ 86.3%

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

      3. associate-/r* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. associate-/l/ 86.3%

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

      2. times-frac 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. associate-*r/ 91.5%

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

       2. associate-/r/ 95.1%

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

   11. Simplified 95.1%

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

   if 1.7e93 < y.im

   1. Initial program 47.1%

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

      1. *-un-lft-identity 47.1%

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

      2. add-sqr-sqrt 47.1%

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

      3. times-frac 47.1%

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

      4. hypot-def 47.1%

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

      5. hypot-def 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in y.re around 0 92.2%

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

      1. neg-mul-1 92.2%

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

      2. +-commutative 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 92.6%

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

   7. Simplified 92.6%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-102}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - t_1\right)\\ \mathbf{elif}\;y.im \leq -2.75 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

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))))
        (t_1 (/ x.im (/ y.im y.re))))
   (if (<= y.im -3.4e+87)
     (* (/ 1.0 (hypot y.re y.im)) (- x.re t_1))
     (if (<= y.im -2.75e-98)
       t_0
       (if (<= y.im 9e-114)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 9e+95) t_0 (* (- t_1 x.re) (/ 1.0 y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 t_1 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.4e+87) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - t_1);
	} else if (y_46_im <= -2.75e-98) {
		tmp = t_0;
	} else if (y_46_im <= 9e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 9e+95) {
		tmp = t_0;
	} else {
		tmp = (t_1 - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_im / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.4e+87) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re - t_1);
	} else if (y_46_im <= -2.75e-98) {
		tmp = t_0;
	} else if (y_46_im <= 9e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 9e+95) {
		tmp = t_0;
	} else {
		tmp = (t_1 - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_im / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -3.4e+87:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re - t_1)
	elif y_46_im <= -2.75e-98:
		tmp = t_0
	elif y_46_im <= 9e-114:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_im <= 9e+95:
		tmp = t_0
	else:
		tmp = (t_1 - x_46_re) * (1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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)))
	t_1 = Float64(x_46_im / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.4e+87)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - t_1));
	elseif (y_46_im <= -2.75e-98)
		tmp = t_0;
	elseif (y_46_im <= 9e-114)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 9e+95)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_1 - x_46_re) * Float64(1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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));
	t_1 = x_46_im / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -3.4e+87)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - t_1);
	elseif (y_46_im <= -2.75e-98)
		tmp = t_0;
	elseif (y_46_im <= 9e-114)
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_im <= 9e+95)
		tmp = t_0;
	else
		tmp = (t_1 - x_46_re) * (1.0 / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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]}, Block[{t$95$1 = N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.4e+87], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.75e-98], t$95$0, If[LessEqual[y$46$im, 9e-114], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9e+95], t$95$0, N[(N[(t$95$1 - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.4000000000000002e87

   1. Initial program 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. add-sqr-sqrt 37.0%

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

      3. times-frac 37.0%

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

      4. hypot-def 37.0%

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

      5. hypot-def 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in y.im around -inf 83.1%

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

      1. mul-1-neg 83.1%

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

      2. unsub-neg 83.1%

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

      3. associate-/l* 84.3%

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

   7. Simplified 84.3%

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

   if -3.4000000000000002e87 < y.im < -2.7499999999999999e-98 or 8.99999999999999937e-114 < y.im < 9.00000000000000033e95

   1. Initial program 86.0%

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

   if -2.7499999999999999e-98 < y.im < 8.99999999999999937e-114

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. associate-/l* 87.7%

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

      5. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

      1. pow2 85.4%

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

      2. associate-*l/ 86.3%

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

      3. associate-/r* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. associate-/l/ 86.3%

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

      2. times-frac 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. associate-*r/ 91.5%

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

       2. associate-/r/ 95.1%

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

   11. Simplified 95.1%

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

   if 9.00000000000000033e95 < y.im

   1. Initial program 47.1%

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

      1. *-un-lft-identity 47.1%

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

      2. add-sqr-sqrt 47.1%

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

      3. times-frac 47.1%

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

      4. hypot-def 47.1%

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

      5. hypot-def 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in y.re around 0 92.2%

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

      1. neg-mul-1 92.2%

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

      2. +-commutative 92.2%

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

      3. unsub-neg 92.2%

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

      4. associate-/l* 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in y.re around 0 91.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{x.im}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.im \leq -2.75 \cdot 10^{-98}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (- (/ x.im (/ y.im y.re)) x.re) (/ 1.0 y.im))))
   (if (<= y.im -3.9e+86)
     t_1
     (if (<= y.im -7.4e-105)
       t_0
       (if (<= y.im 6.2e-114)
         (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))
         (if (<= y.im 2.5e+95) 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_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	double tmp;
	if (y_46_im <= -3.9e+86) {
		tmp = t_1;
	} else if (y_46_im <= -7.4e-105) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 2.5e+95) {
		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_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((x_46im / (y_46im / y_46re)) - x_46re) * (1.0d0 / y_46im)
    if (y_46im <= (-3.9d+86)) then
        tmp = t_1
    else if (y_46im <= (-7.4d-105)) then
        tmp = t_0
    else if (y_46im <= 6.2d-114) then
        tmp = (x_46im / y_46re) - ((x_46re / (y_46re / y_46im)) / y_46re)
    else if (y_46im <= 2.5d+95) 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_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	double tmp;
	if (y_46_im <= -3.9e+86) {
		tmp = t_1;
	} else if (y_46_im <= -7.4e-105) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-114) {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	} else if (y_46_im <= 2.5e+95) {
		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_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im)
	tmp = 0
	if y_46_im <= -3.9e+86:
		tmp = t_1
	elif y_46_im <= -7.4e-105:
		tmp = t_0
	elif y_46_im <= 6.2e-114:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	elif y_46_im <= 2.5e+95:
		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 * 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)))
	t_1 = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) * Float64(1.0 / y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.9e+86)
		tmp = t_1;
	elseif (y_46_im <= -7.4e-105)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-114)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	elseif (y_46_im <= 2.5e+95)
		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_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -3.9e+86)
		tmp = t_1;
	elseif (y_46_im <= -7.4e-105)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-114)
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	elseif (y_46_im <= 2.5e+95)
		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 * 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]}, Block[{t$95$1 = N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.9e+86], t$95$1, If[LessEqual[y$46$im, -7.4e-105], t$95$0, If[LessEqual[y$46$im, 6.2e-114], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.5e+95], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.9000000000000002e86 or 2.50000000000000012e95 < y.im

   1. Initial program 41.9%

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

      1. *-un-lft-identity 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. times-frac 41.9%

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

      4. hypot-def 41.9%

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

      5. hypot-def 64.5%

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

   4. Applied egg-rr 64.5%

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

   5. Taylor expanded in y.re around 0 54.9%

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

      1. neg-mul-1 54.9%

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

      2. +-commutative 54.9%

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

      3. unsub-neg 54.9%

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

      4. associate-/l* 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in y.re around 0 87.9%

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

   if -3.9000000000000002e86 < y.im < -7.40000000000000017e-105 or 6.2e-114 < y.im < 2.50000000000000012e95

   1. Initial program 86.0%

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

   if -7.40000000000000017e-105 < y.im < 6.2e-114

   1. Initial program 71.4%

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

   3. Taylor expanded in y.re around inf 86.3%

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

      1. +-commutative 86.3%

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

      2. mul-1-neg 86.3%

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

      3. unsub-neg 86.3%

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

      4. associate-/l* 87.7%

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

      5. associate-/r/ 85.4%

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

   5. Simplified 85.4%

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

      1. pow2 85.4%

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

      2. associate-*l/ 86.3%

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

      3. associate-/r* 94.7%

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

   7. Applied egg-rr 94.7%

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

      1. associate-/l/ 86.3%

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

      2. times-frac 91.5%

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

   9. Applied egg-rr 91.5%

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

       1. associate-*r/ 91.5%

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

       2. associate-/r/ 95.1%

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

   11. Simplified 95.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{+86}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+31} \lor \neg \left(y.re \leq 0.6\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e+31) (not (<= y.re 0.6)))
   (/ x.im y.re)
   (* (- (/ x.im (/ y.im y.re)) x.re) (/ 1.0 y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6e+31) || !(y_46_re <= 0.6)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6d+31)) .or. (.not. (y_46re <= 0.6d0))) then
        tmp = x_46im / y_46re
    else
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) * (1.0d0 / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6e+31) || !(y_46_re <= 0.6)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6e+31) or not (y_46_re <= 0.6):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6e+31) || !(y_46_re <= 0.6))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) * Float64(1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6e+31) || ~((y_46_re <= 0.6)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -6e+31], N[Not[LessEqual[y$46$re, 0.6]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.99999999999999978e31 or 0.599999999999999978 < y.re

   1. Initial program 51.4%

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

   3. Taylor expanded in y.re around inf 69.3%

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

   if -5.99999999999999978e31 < y.re < 0.599999999999999978

   1. Initial program 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. add-sqr-sqrt 74.0%

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

      3. times-frac 74.0%

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

      4. hypot-def 74.0%

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

      5. hypot-def 87.0%

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

   4. Applied egg-rr 87.0%

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

   5. Taylor expanded in y.re around 0 45.3%

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

      1. neg-mul-1 45.3%

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

      2. +-commutative 45.3%

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

      3. unsub-neg 45.3%

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

      4. associate-/l* 44.7%

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

   7. Simplified 44.7%

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

   8. Taylor expanded in y.re around 0 81.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+31} \lor \neg \left(y.re \leq 0.6\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-85} \lor \neg \left(y.im \leq 7.8 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.2e-85) (not (<= y.im 7.8e-18)))
   (* (- (/ x.im (/ y.im y.re)) x.re) (/ 1.0 y.im))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e-85) || !(y_46_im <= 7.8e-18)) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.2d-85)) .or. (.not. (y_46im <= 7.8d-18))) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) * (1.0d0 / y_46im)
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e-85) || !(y_46_im <= 7.8e-18)) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.2e-85) or not (y_46_im <= 7.8e-18):
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.2e-85) || !(y_46_im <= 7.8e-18))
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.2e-85) || ~((y_46_im <= 7.8e-18)))
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.2e-85], N[Not[LessEqual[y$46$im, 7.8e-18]], $MachinePrecision]], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.2e-85 or 7.8000000000000001e-18 < y.im

   1. Initial program 57.3%

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

      1. *-un-lft-identity 57.3%

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

      2. add-sqr-sqrt 57.3%

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

      3. times-frac 57.3%

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

      4. hypot-def 57.3%

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

      5. hypot-def 72.7%

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in y.re around 0 40.9%

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

      1. neg-mul-1 40.9%

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

      2. +-commutative 40.9%

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

      3. unsub-neg 40.9%

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

      4. associate-/l* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in y.re around 0 77.6%

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

   if -4.2e-85 < y.im < 7.8000000000000001e-18

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf 80.3%

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

      1. +-commutative 80.3%

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

      2. mul-1-neg 80.3%

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

      3. unsub-neg 80.3%

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

      4. associate-/l* 81.4%

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

      5. associate-/r/ 79.6%

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

   5. Simplified 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. pow2 79.6%

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

      3. times-frac 81.6%

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

   7. Applied egg-rr 81.6%

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

      1. associate-*l/ 81.6%

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

      2. *-un-lft-identity 81.6%

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

   9. Applied egg-rr 81.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-85} \lor \neg \left(y.im \leq 7.8 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-84} \lor \neg \left(y.im \leq 6.2 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.3e-84) (not (<= y.im 6.2e-13)))
   (* (- (/ x.im (/ y.im y.re)) x.re) (/ 1.0 y.im))
   (- (/ x.im y.re) (/ (/ x.re (/ y.re y.im)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.3e-84) || !(y_46_im <= 6.2e-13)) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.3d-84)) .or. (.not. (y_46im <= 6.2d-13))) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) * (1.0d0 / y_46im)
    else
        tmp = (x_46im / y_46re) - ((x_46re / (y_46re / y_46im)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.3e-84) || !(y_46_im <= 6.2e-13)) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.3e-84) or not (y_46_im <= 6.2e-13):
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.3e-84) || !(y_46_im <= 6.2e-13))
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.3e-84) || ~((y_46_im <= 6.2e-13)))
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) * (1.0 / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / (y_46_re / y_46_im)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.3e-84], N[Not[LessEqual[y$46$im, 6.2e-13]], $MachinePrecision]], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.3e-84 or 6.1999999999999998e-13 < y.im

   1. Initial program 57.3%

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

      1. *-un-lft-identity 57.3%

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

      2. add-sqr-sqrt 57.3%

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

      3. times-frac 57.3%

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

      4. hypot-def 57.3%

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

      5. hypot-def 72.7%

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in y.re around 0 40.9%

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

      1. neg-mul-1 40.9%

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

      2. +-commutative 40.9%

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

      3. unsub-neg 40.9%

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

      4. associate-/l* 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in y.re around 0 77.6%

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

   if -1.3e-84 < y.im < 6.1999999999999998e-13

   1. Initial program 76.2%

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

   3. Taylor expanded in y.re around inf 80.3%

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

      1. +-commutative 80.3%

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

      2. mul-1-neg 80.3%

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

      3. unsub-neg 80.3%

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

      4. associate-/l* 81.4%

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

      5. associate-/r/ 79.6%

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

   5. Simplified 79.6%

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

      1. pow2 79.6%

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

      2. associate-*l/ 80.3%

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

      3. associate-/r* 87.0%

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

   7. Applied egg-rr 87.0%

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

      1. associate-/l/ 80.3%

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

      2. times-frac 84.5%

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

   9. Applied egg-rr 84.5%

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

       1. associate-*r/ 84.4%

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

       2. associate-/r/ 87.3%

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

   11. Simplified 87.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-84} \lor \neg \left(y.im \leq 6.2 \cdot 10^{-13}\right):\\ \;\;\;\;\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-50} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.05e-50) (not (<= y.im 3.8e-55)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.05e-50) || !(y_46_im <= 3.8e-55)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.05d-50)) .or. (.not. (y_46im <= 3.8d-55))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.05e-50) || !(y_46_im <= 3.8e-55)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.05e-50) or not (y_46_im <= 3.8e-55):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.05e-50) || !(y_46_im <= 3.8e-55))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.05e-50) || ~((y_46_im <= 3.8e-55)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.05e-50], N[Not[LessEqual[y$46$im, 3.8e-55]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.04999999999999993e-50 or 3.7999999999999997e-55 < y.im

   1. Initial program 57.7%

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

   3. Taylor expanded in y.re around 0 72.0%

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

      1. associate-*r/ 72.0%

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

      2. neg-mul-1 72.0%

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

   5. Simplified 72.0%

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

   if -2.04999999999999993e-50 < y.im < 3.7999999999999997e-55

   1. Initial program 74.9%

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

   3. Taylor expanded in y.re around inf 65.0%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-50} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+158} \lor \neg \left(y.im \leq 9 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.4e+158) (not (<= y.im 9e+166)))
   (/ x.re y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.4e+158) || !(y_46_im <= 9e+166)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.4d+158)) .or. (.not. (y_46im <= 9d+166))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.4e+158) || !(y_46_im <= 9e+166)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.4e+158) or not (y_46_im <= 9e+166):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.4e+158) || !(y_46_im <= 9e+166))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.4e+158) || ~((y_46_im <= 9e+166)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.4e+158], N[Not[LessEqual[y$46$im, 9e+166]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.40000000000000021e158 or 9.00000000000000061e166 < y.im

   1. Initial program 31.6%

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

      1. *-un-lft-identity 31.6%

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

      2. add-sqr-sqrt 31.6%

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

      3. times-frac 31.6%

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

      4. hypot-def 31.6%

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

      5. hypot-def 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in y.re around 0 60.4%

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

      1. neg-mul-1 60.4%

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

      2. +-commutative 60.4%

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

      3. unsub-neg 60.4%

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

      4. associate-/l* 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in y.im around -inf 32.1%

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

   if -7.40000000000000021e158 < y.im < 9.00000000000000061e166

   1. Initial program 75.5%

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

   3. Taylor expanded in y.re around inf 48.9%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+158} \lor \neg \left(y.im \leq 9 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 9.1% 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 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. add-sqr-sqrt 64.5%

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

   3. times-frac 64.5%

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

   4. hypot-def 64.5%

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

   5. hypot-def 78.0%

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

4. Applied egg-rr 78.0%

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

5. Taylor expanded in y.re around 0 33.6%

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

   1. neg-mul-1 33.6%

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

   2. +-commutative 33.6%

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

   3. unsub-neg 33.6%

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

   4. associate-/l* 33.6%

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

7. Simplified 33.6%

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

8. Taylor expanded in y.re around inf 9.0%

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

9. Final simplification 9.0%

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

Alternative 14: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in y.re around inf 38.8%

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

4. Final simplification 38.8%

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

Reproduce

herbie shell --seed 2024013 
(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))))