_divideComplex, imaginary part

Percentage Accurate: 62.1% → 98.2%

Time: 16.8s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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[(N[(x$46$re * N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision])), $MachinePrecision]), $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. div-sub 60.5%

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

   2. *-commutative 60.5%

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

   3. add-sqr-sqrt 60.5%

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

   4. times-frac 63.8%

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

   5. fma-neg 63.8%

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

   6. hypot-define 63.8%

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

   7. hypot-define 77.2%

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

   8. associate-/l* 80.6%

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

   9. add-sqr-sqrt 80.6%

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

   10. pow2 80.6%

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

   11. hypot-define 80.6%

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

4. Applied egg-rr 80.6%

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

   1. *-un-lft-identity 80.6%

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

   2. unpow2 80.6%

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

   3. times-frac 97.2%

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

   4. hypot-undefine 80.6%

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

   5. +-commutative 80.6%

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

   6. hypot-undefine 97.2%

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

   7. hypot-undefine 80.6%

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

   8. +-commutative 80.6%

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

   9. hypot-undefine 97.2%

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

6. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. *-lft-identity 97.2%

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

8. Simplified 97.2%

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

   1. associate-*r/ 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

    \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.re \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
12. Add Preprocessing

Alternative 2: 90.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+141} \lor \neg \left(y.im \leq 7 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (hypot y.re y.im))) (t_1 (/ y.re (hypot y.re y.im))))
   (if (or (<= y.im -2.6e+141) (not (<= y.im 7e+67)))
     (fma t_1 t_0 (* x.re (/ -1.0 y.im)))
     (fma t_1 t_0 (* x.re (/ y.im (- (pow (hypot y.re y.im) 2.0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_im / hypot(y_46_re, y_46_im);
	double t_1 = y_46_re / hypot(y_46_re, y_46_im);
	double tmp;
	if ((y_46_im <= -2.6e+141) || !(y_46_im <= 7e+67)) {
		tmp = fma(t_1, t_0, (x_46_re * (-1.0 / y_46_im)));
	} else {
		tmp = fma(t_1, t_0, (x_46_re * (y_46_im / -pow(hypot(y_46_re, y_46_im), 2.0))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	t_1 = Float64(y_46_re / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if ((y_46_im <= -2.6e+141) || !(y_46_im <= 7e+67))
		tmp = fma(t_1, t_0, Float64(x_46_re * Float64(-1.0 / y_46_im)));
	else
		tmp = fma(t_1, t_0, Float64(x_46_re * Float64(y_46_im / Float64(-(hypot(y_46_re, y_46_im) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -2.6e+141], N[Not[LessEqual[y$46$im, 7e+67]], $MachinePrecision]], N[(t$95$1 * t$95$0 + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0 + N[(x$46$re * N[(y$46$im / (-N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5999999999999999e141 or 7e67 < y.im

   1. Initial program 47.0%

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

      1. div-sub 47.0%

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

      2. *-commutative 47.0%

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

      3. add-sqr-sqrt 47.0%

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

      4. times-frac 48.3%

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

      5. fma-neg 48.3%

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

      6. hypot-define 48.3%

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

      7. hypot-define 58.8%

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

      8. associate-/l* 62.6%

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

      9. add-sqr-sqrt 62.6%

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

      10. pow2 62.6%

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

      11. hypot-define 62.6%

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

   4. Applied egg-rr 62.6%

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

   5. Taylor expanded in y.im around inf 96.4%

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

   if -2.5999999999999999e141 < y.im < 7e67

   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. Step-by-step derivation

      1. div-sub 68.6%

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

      2. *-commutative 68.6%

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

      3. add-sqr-sqrt 68.6%

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

      4. times-frac 73.1%

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

      5. fma-neg 73.1%

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

      6. hypot-define 73.1%

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

      7. hypot-define 88.3%

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

      8. associate-/l* 91.4%

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

      9. add-sqr-sqrt 91.4%

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

      10. pow2 91.4%

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

      11. hypot-define 91.4%

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

   4. Applied egg-rr 91.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+141} \lor \neg \left(y.im \leq 7 \cdot 10^{+67}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $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. div-sub 60.5%

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

   2. *-commutative 60.5%

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

   3. add-sqr-sqrt 60.5%

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

   4. times-frac 63.8%

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

   5. fma-neg 63.8%

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

   6. hypot-define 63.8%

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

   7. hypot-define 77.2%

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

   8. associate-/l* 80.6%

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

   9. add-sqr-sqrt 80.6%

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

   10. pow2 80.6%

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

   11. hypot-define 80.6%

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

4. Applied egg-rr 80.6%

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

   1. *-un-lft-identity 80.6%

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

   2. unpow2 80.6%

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

   3. times-frac 97.2%

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

   4. hypot-undefine 80.6%

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

   5. +-commutative 80.6%

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

   6. hypot-undefine 97.2%

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

   7. hypot-undefine 80.6%

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

   8. +-commutative 80.6%

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

   9. hypot-undefine 97.2%

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

6. Applied egg-rr 97.2%

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

   1. associate-*l/ 97.2%

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

   2. *-lft-identity 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

   \[\leadsto \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{-\mathsf{hypot}\left(y.im, y.re\right)}\right) \]
10. Add Preprocessing

Alternative 4: 89.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+285}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+285)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (* x.re (/ -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))) <= 1e+285) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re * (-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))) <= 1e+285)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(-1.0 / y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+285], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.9999999999999998e284

   1. Initial program 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. times-frac 82.0%

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

      4. hypot-define 82.0%

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

      5. fma-neg 82.0%

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

      6. distribute-rgt-neg-in 82.0%

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

      7. hypot-define 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. associate-*l/ 96.9%

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

      2. *-un-lft-identity 96.9%

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

      3. distribute-rgt-neg-out 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-neg 96.9%

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

      6. *-commutative 96.9%

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

      7. *-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 9.9999999999999998e284 < (/.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 13.0%

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

      1. div-sub 6.8%

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

      2. *-commutative 6.8%

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

      3. add-sqr-sqrt 6.8%

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

      4. times-frac 16.9%

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

      5. fma-neg 16.9%

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

      6. hypot-define 16.9%

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

      7. hypot-define 52.5%

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

      8. associate-/l* 61.4%

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

      9. add-sqr-sqrt 61.4%

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

      10. pow2 61.4%

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

      11. hypot-define 61.4%

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

   4. Applied egg-rr 61.4%

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

   5. Taylor expanded in y.im around inf 71.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+285}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{-1}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+307)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (- (/ 1.0 (* y.im (/ (/ y.im y.re) x.im))) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+307) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+307) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (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))) <= 1e+307:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+307)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(1.0 / Float64(y_46_im * Float64(Float64(y_46_im / y_46_re) / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (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))) <= 1e+307)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+307], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.99999999999999986e306

   1. Initial program 82.2%

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

      1. *-un-lft-identity 82.2%

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

      2. add-sqr-sqrt 82.2%

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

      3. times-frac 82.3%

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

      4. hypot-define 82.3%

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

      5. fma-neg 82.3%

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

      6. distribute-rgt-neg-in 82.3%

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

      7. hypot-define 96.7%

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

   4. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.9%

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

      2. *-un-lft-identity 96.9%

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

      3. distribute-rgt-neg-out 96.9%

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

      4. *-commutative 96.9%

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

      5. fma-neg 96.9%

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

      6. *-commutative 96.9%

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

      7. *-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 9.99999999999999986e306 < (/.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 8.8%

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

   3. Taylor expanded in y.re around 0 40.2%

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

      1. +-commutative 40.2%

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

      2. mul-1-neg 40.2%

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

      3. unsub-neg 40.2%

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

      4. associate-/l* 45.6%

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

   5. Simplified 45.6%

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

      1. *-un-lft-identity 45.6%

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

      2. pow2 45.6%

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

      3. times-frac 54.5%

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

   7. Applied egg-rr 54.5%

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

      1. associate-*r* 56.5%

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

      2. clear-num 56.4%

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

      3. un-div-inv 56.5%

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

      4. un-div-inv 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. clear-num 56.5%

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

       2. inv-pow 56.5%

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

   11. Applied egg-rr 56.5%

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

       1. unpow-1 56.5%

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

       2. associate-/r/ 56.5%

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

   13. Simplified 56.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- (/ 1.0 (* y.im (/ (/ y.im y.re) x.im))) (/ x.re y.im))))
   (if (<= y.im -4.9e+93)
     t_1
     (if (<= y.im -1.95e-26)
       t_0
       (if (<= y.im 1.25e-140)
         (- (/ x.im y.re) (* x.re (* y.im (pow y.re -2.0))))
         (if (<= y.im 7e+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.9e+93) {
		tmp = t_1;
	} else if (y_46_im <= -1.95e-26) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-140) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im * pow(y_46_re, -2.0)));
	} else if (y_46_im <= 7e+67) {
		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 = (1.0d0 / (y_46im * ((y_46im / y_46re) / x_46im))) - (x_46re / y_46im)
    if (y_46im <= (-4.9d+93)) then
        tmp = t_1
    else if (y_46im <= (-1.95d-26)) then
        tmp = t_0
    else if (y_46im <= 1.25d-140) then
        tmp = (x_46im / y_46re) - (x_46re * (y_46im * (y_46re ** (-2.0d0))))
    else if (y_46im <= 7d+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.9e+93) {
		tmp = t_1;
	} else if (y_46_im <= -1.95e-26) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-140) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im * Math.pow(y_46_re, -2.0)));
	} else if (y_46_im <= 7e+67) {
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -4.9e+93:
		tmp = t_1
	elif y_46_im <= -1.95e-26:
		tmp = t_0
	elif y_46_im <= 1.25e-140:
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im * math.pow(y_46_re, -2.0)))
	elif y_46_im <= 7e+67:
		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(1.0 / Float64(y_46_im * Float64(Float64(y_46_im / y_46_re) / x_46_im))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.9e+93)
		tmp = t_1;
	elseif (y_46_im <= -1.95e-26)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-140)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im * (y_46_re ^ -2.0))));
	elseif (y_46_im <= 7e+67)
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4.9e+93)
		tmp = t_1;
	elseif (y_46_im <= -1.95e-26)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-140)
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im * (y_46_re ^ -2.0)));
	elseif (y_46_im <= 7e+67)
		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[(1.0 / N[(y$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.9e+93], t$95$1, If[LessEqual[y$46$im, -1.95e-26], t$95$0, If[LessEqual[y$46$im, 1.25e-140], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im * N[Power[y$46$re, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e+67], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.89999999999999969e93 or 7e67 < y.im

   1. Initial program 46.6%

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

   3. Taylor expanded in y.re around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. mul-1-neg 80.8%

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

      3. unsub-neg 80.8%

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

      4. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. pow2 82.9%

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

      3. times-frac 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. associate-*r* 89.9%

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

      2. clear-num 89.9%

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

      3. un-div-inv 89.9%

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

      4. un-div-inv 89.9%

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

   9. Applied egg-rr 89.9%

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

       1. clear-num 89.9%

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

       2. inv-pow 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. unpow-1 89.9%

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

       2. associate-/r/ 89.9%

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

   13. Simplified 89.9%

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

   if -4.89999999999999969e93 < y.im < -1.94999999999999993e-26 or 1.25000000000000004e-140 < y.im < 7e67

   1. Initial program 81.9%

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

   if -1.94999999999999993e-26 < y.im < 1.25000000000000004e-140

   1. Initial program 75.1%

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

   3. Taylor expanded in y.re around inf 85.5%

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

      1. +-commutative 85.5%

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

      2. mul-1-neg 85.5%

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

      3. unsub-neg 85.5%

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

      4. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

      1. div-inv 84.9%

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

      2. pow-flip 84.9%

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

      3. metadata-eval 84.9%

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

   7. Applied egg-rr 84.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \left(y.im \cdot {y.re}^{-2}\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- (/ 1.0 (* y.im (/ (/ y.im y.re) x.im))) (/ x.re y.im))))
   (if (<= y.im -1.65e+93)
     t_1
     (if (<= y.im -4.8e-29)
       t_0
       (if (<= y.im 2.3e-141)
         (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
         (if (<= y.im 6.5e+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.65e+93) {
		tmp = t_1;
	} else if (y_46_im <= -4.8e-29) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-141) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 6.5e+67) {
		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 = (1.0d0 / (y_46im * ((y_46im / y_46re) / x_46im))) - (x_46re / y_46im)
    if (y_46im <= (-1.65d+93)) then
        tmp = t_1
    else if (y_46im <= (-4.8d-29)) then
        tmp = t_0
    else if (y_46im <= 2.3d-141) then
        tmp = (x_46im / y_46re) - (x_46re * (y_46im / (y_46re ** 2.0d0)))
    else if (y_46im <= 6.5d+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.65e+93) {
		tmp = t_1;
	} else if (y_46_im <= -4.8e-29) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-141) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 6.5e+67) {
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.65e+93:
		tmp = t_1
	elif y_46_im <= -4.8e-29:
		tmp = t_0
	elif y_46_im <= 2.3e-141:
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 6.5e+67:
		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(1.0 / Float64(y_46_im * Float64(Float64(y_46_im / y_46_re) / x_46_im))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.65e+93)
		tmp = t_1;
	elseif (y_46_im <= -4.8e-29)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-141)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 6.5e+67)
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.65e+93)
		tmp = t_1;
	elseif (y_46_im <= -4.8e-29)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-141)
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 6.5e+67)
		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[(1.0 / N[(y$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+93], t$95$1, If[LessEqual[y$46$im, -4.8e-29], t$95$0, If[LessEqual[y$46$im, 2.3e-141], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.5e+67], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65000000000000004e93 or 6.4999999999999995e67 < y.im

   1. Initial program 46.6%

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

   3. Taylor expanded in y.re around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. mul-1-neg 80.8%

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

      3. unsub-neg 80.8%

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

      4. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. pow2 82.9%

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

      3. times-frac 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. associate-*r* 89.9%

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

      2. clear-num 89.9%

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

      3. un-div-inv 89.9%

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

      4. un-div-inv 89.9%

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

   9. Applied egg-rr 89.9%

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

       1. clear-num 89.9%

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

       2. inv-pow 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. unpow-1 89.9%

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

       2. associate-/r/ 89.9%

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

   13. Simplified 89.9%

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

   if -1.65000000000000004e93 < y.im < -4.79999999999999984e-29 or 2.29999999999999995e-141 < y.im < 6.4999999999999995e67

   1. Initial program 81.9%

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

   if -4.79999999999999984e-29 < y.im < 2.29999999999999995e-141

   1. Initial program 75.1%

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

   3. Taylor expanded in y.re around inf 85.5%

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

      1. +-commutative 85.5%

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

      2. mul-1-neg 85.5%

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

      3. unsub-neg 85.5%

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

      4. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+93}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.2e+38)
   (* (/ 1.0 (hypot y.re y.im)) (- x.re (* x.im (/ y.re y.im))))
   (if (<= y.im 2.6e-139)
     (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))
     (if (<= y.im 6.4e+67)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (- (/ 1.0 (* y.im (/ (/ y.im y.re) x.im))) (/ x.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.2e+38) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - (x_46_im * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.6e-139) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 6.4e+67) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.2e+38) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re - (x_46_im * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.6e-139) {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 6.4e+67) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.2e+38:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re - (x_46_im * (y_46_re / y_46_im)))
	elif y_46_im <= 2.6e-139:
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 6.4e+67:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.2e+38)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - Float64(x_46_im * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.6e-139)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 6.4e+67)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(1.0 / Float64(y_46_im * Float64(Float64(y_46_im / y_46_re) / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.2e+38)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - (x_46_im * (y_46_re / y_46_im)));
	elseif (y_46_im <= 2.6e-139)
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 6.4e+67)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.2e+38], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.6e-139], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.4e+67], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.20000000000000006e38

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.4%

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

      4. hypot-define 46.4%

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

      5. fma-neg 46.4%

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

      6. distribute-rgt-neg-in 46.4%

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

      7. hypot-define 66.5%

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in y.im around -inf 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

      3. associate-/l* 83.4%

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

   7. Simplified 83.4%

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

   if -2.20000000000000006e38 < y.im < 2.5999999999999998e-139

   1. Initial program 76.8%

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

   3. Taylor expanded in y.re around inf 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if 2.5999999999999998e-139 < y.im < 6.39999999999999965e67

   1. Initial program 81.1%

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

   if 6.39999999999999965e67 < y.im

   1. Initial program 50.1%

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

   3. Taylor expanded in y.re around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. mul-1-neg 84.7%

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

      3. unsub-neg 84.7%

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

      4. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

      1. *-un-lft-identity 87.1%

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

      2. pow2 87.1%

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

      3. times-frac 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. associate-*r* 94.3%

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

      2. clear-num 94.3%

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

      3. un-div-inv 94.3%

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

      4. un-div-inv 94.3%

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

   9. Applied egg-rr 94.3%

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

       1. clear-num 94.3%

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

       2. inv-pow 94.3%

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

   11. Applied egg-rr 94.3%

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

       1. unpow-1 94.3%

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

       2. associate-/r/ 94.3%

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

   13. Simplified 94.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - x.im \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{+67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- (/ 1.0 (* y.im (/ (/ y.im y.re) x.im))) (/ x.re y.im))))
   (if (<= y.im -2.35e+92)
     t_1
     (if (<= y.im -9e-193)
       t_0
       (if (<= y.im 1.75e-163)
         (/ x.im y.re)
         (if (<= y.im 1.15e+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.35e+92) {
		tmp = t_1;
	} else if (y_46_im <= -9e-193) {
		tmp = t_0;
	} else if (y_46_im <= 1.75e-163) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.15e+67) {
		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 = (1.0d0 / (y_46im * ((y_46im / y_46re) / x_46im))) - (x_46re / y_46im)
    if (y_46im <= (-2.35d+92)) then
        tmp = t_1
    else if (y_46im <= (-9d-193)) then
        tmp = t_0
    else if (y_46im <= 1.75d-163) then
        tmp = x_46im / y_46re
    else if (y_46im <= 1.15d+67) 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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.35e+92) {
		tmp = t_1;
	} else if (y_46_im <= -9e-193) {
		tmp = t_0;
	} else if (y_46_im <= 1.75e-163) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.15e+67) {
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.35e+92:
		tmp = t_1
	elif y_46_im <= -9e-193:
		tmp = t_0
	elif y_46_im <= 1.75e-163:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1.15e+67:
		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(1.0 / Float64(y_46_im * Float64(Float64(y_46_im / y_46_re) / x_46_im))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.35e+92)
		tmp = t_1;
	elseif (y_46_im <= -9e-193)
		tmp = t_0;
	elseif (y_46_im <= 1.75e-163)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.15e+67)
		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 = (1.0 / (y_46_im * ((y_46_im / y_46_re) / x_46_im))) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.35e+92)
		tmp = t_1;
	elseif (y_46_im <= -9e-193)
		tmp = t_0;
	elseif (y_46_im <= 1.75e-163)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1.15e+67)
		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[(1.0 / N[(y$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.35e+92], t$95$1, If[LessEqual[y$46$im, -9e-193], t$95$0, If[LessEqual[y$46$im, 1.75e-163], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.15e+67], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.35e92 or 1.1499999999999999e67 < y.im

   1. Initial program 46.6%

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

   3. Taylor expanded in y.re around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. mul-1-neg 80.8%

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

      3. unsub-neg 80.8%

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

      4. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. pow2 82.9%

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

      3. times-frac 84.4%

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

   7. Applied egg-rr 84.4%

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

      1. associate-*r* 89.9%

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

      2. clear-num 89.9%

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

      3. un-div-inv 89.9%

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

      4. un-div-inv 89.9%

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

   9. Applied egg-rr 89.9%

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

       1. clear-num 89.9%

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

       2. inv-pow 89.9%

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

   11. Applied egg-rr 89.9%

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

       1. unpow-1 89.9%

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

       2. associate-/r/ 89.9%

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

   13. Simplified 89.9%

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

   if -2.35e92 < y.im < -8.9999999999999997e-193 or 1.75000000000000014e-163 < y.im < 1.1499999999999999e67

   1. Initial program 81.3%

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

   if -8.9999999999999997e-193 < y.im < 1.75000000000000014e-163

   1. Initial program 72.8%

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

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

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.35 \cdot 10^{+92}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-193}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im \cdot \frac{\frac{y.im}{y.re}}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e+143)
   (/ x.im y.re)
   (if (<= y.re -1.8e-30)
     (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 5.5e+58)
       (- (/ (/ (* y.re x.im) y.im) y.im) (/ x.re y.im))
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e+143) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.8e-30) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5.5e+58) {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.1d+143)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-1.8d-30)) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 5.5d+58) then
        tmp = (((y_46re * x_46im) / y_46im) / y_46im) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e+143) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.8e-30) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5.5e+58) {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.1e+143:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -1.8e-30:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 5.5e+58:
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.1e+143)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.8e-30)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 5.5e+58)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.1e+143)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -1.8e-30)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 5.5e+58)
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.1e+143], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.8e-30], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.5e+58], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.09999999999999988e143 or 5.4999999999999999e58 < y.re

   1. Initial program 39.3%

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

   3. Taylor expanded in y.re around inf 78.4%

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

   if -2.09999999999999988e143 < y.re < -1.8000000000000002e-30

   1. Initial program 78.7%

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

   3. Taylor expanded in x.im around inf 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   if -1.8000000000000002e-30 < y.re < 5.4999999999999999e58

   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 0 74.1%

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

      1. +-commutative 74.1%

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

      2. mul-1-neg 74.1%

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

      3. unsub-neg 74.1%

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

      4. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. associate-*r/ 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-/r* 79.5%

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

   7. Applied egg-rr 79.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+27} \lor \neg \left(y.im \leq 3.35 \cdot 10^{-133}\right):\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.16e+27) (not (<= y.im 3.35e-133)))
   (- (* x.im (/ (/ y.re y.im) y.im)) (/ 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 <= -1.16e+27) || !(y_46_im <= 3.35e-133)) {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (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 <= (-1.16d+27)) .or. (.not. (y_46im <= 3.35d-133))) then
        tmp = (x_46im * ((y_46re / y_46im) / y_46im)) - (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 <= -1.16e+27) || !(y_46_im <= 3.35e-133)) {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (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 <= -1.16e+27) or not (y_46_im <= 3.35e-133):
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (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 <= -1.16e+27) || !(y_46_im <= 3.35e-133))
		tmp = Float64(Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) / y_46_im)) - 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 <= -1.16e+27) || ~((y_46_im <= 3.35e-133)))
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (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, -1.16e+27], N[Not[LessEqual[y$46$im, 3.35e-133]], $MachinePrecision]], N[(N[(x$46$im * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.16e27 or 3.3500000000000001e-133 < y.im

   1. Initial program 55.6%

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

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

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

      1. +-commutative 72.6%

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

      2. mul-1-neg 72.6%

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

      3. unsub-neg 72.6%

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

      4. associate-/l* 73.6%

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

   5. Simplified 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. pow2 73.6%

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

      3. times-frac 74.6%

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

   7. Applied egg-rr 74.6%

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

      1. associate-*l/ 74.6%

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

      2. *-lft-identity 74.6%

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

   9. Simplified 74.6%

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

   if -1.16e27 < y.im < 3.3500000000000001e-133

   1. Initial program 76.8%

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

   3. Taylor expanded in y.re around inf 76.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+27} \lor \neg \left(y.im \leq 3.35 \cdot 10^{-133}\right):\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+29} \lor \neg \left(y.im \leq 3.6 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\frac{x.im}{y.im}}{\frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.25e+29) (not (<= y.im 3.6e-38)))
   (- (/ (/ x.im y.im) (/ y.im y.re)) (/ 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 <= -1.25e+29) || !(y_46_im <= 3.6e-38)) {
		tmp = ((x_46_im / y_46_im) / (y_46_im / y_46_re)) - (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 <= (-1.25d+29)) .or. (.not. (y_46im <= 3.6d-38))) then
        tmp = ((x_46im / y_46im) / (y_46im / y_46re)) - (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 <= -1.25e+29) || !(y_46_im <= 3.6e-38)) {
		tmp = ((x_46_im / y_46_im) / (y_46_im / y_46_re)) - (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 <= -1.25e+29) or not (y_46_im <= 3.6e-38):
		tmp = ((x_46_im / y_46_im) / (y_46_im / y_46_re)) - (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 <= -1.25e+29) || !(y_46_im <= 3.6e-38))
		tmp = Float64(Float64(Float64(x_46_im / y_46_im) / Float64(y_46_im / y_46_re)) - 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 <= -1.25e+29) || ~((y_46_im <= 3.6e-38)))
		tmp = ((x_46_im / y_46_im) / (y_46_im / y_46_re)) - (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, -1.25e+29], N[Not[LessEqual[y$46$im, 3.6e-38]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.25e29 or 3.6000000000000001e-38 < y.im

   1. Initial program 53.9%

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

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

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. *-un-lft-identity 76.4%

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

      2. pow2 76.4%

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

      3. times-frac 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. associate-*r* 82.3%

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

      2. clear-num 81.9%

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

      3. un-div-inv 81.9%

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

      4. un-div-inv 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.25e29 < y.im < 3.6000000000000001e-38

   1. Initial program 76.0%

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

   3. Taylor expanded in y.re around inf 72.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+29} \lor \neg \left(y.im \leq 3.6 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\frac{x.im}{y.im}}{\frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-31} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -7.2e-31) (not (<= y.re 2.6e+57)))
   (/ x.im y.re)
   (- (/ (/ (* y.re x.im) y.im) y.im) (/ x.re y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -7.2e-31) || !(y_46_re <= 2.6e+57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-7.2d-31)) .or. (.not. (y_46re <= 2.6d+57))) then
        tmp = x_46im / y_46re
    else
        tmp = (((y_46re * x_46im) / y_46im) / y_46im) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -7.2e-31) || !(y_46_re <= 2.6e+57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -7.2e-31) or not (y_46_re <= 2.6e+57):
		tmp = x_46_im / y_46_re
	else:
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -7.2e-31) || !(y_46_re <= 2.6e+57))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -7.2e-31) || ~((y_46_re <= 2.6e+57)))
		tmp = x_46_im / y_46_re;
	else
		tmp = (((y_46_re * x_46_im) / y_46_im) / y_46_im) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -7.2e-31], N[Not[LessEqual[y$46$re, 2.6e+57]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.20000000000000007e-31 or 2.6e57 < y.re

   1. Initial program 52.3%

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

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

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

   if -7.20000000000000007e-31 < y.re < 2.6e57

   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 0 74.1%

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

      1. +-commutative 74.1%

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

      2. mul-1-neg 74.1%

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

      3. unsub-neg 74.1%

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

      4. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. associate-*r/ 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-/r* 79.5%

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

   7. Applied egg-rr 79.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{-31} \lor \neg \left(y.re \leq 2.6 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+101} \lor \neg \left(y.im \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.5e+101) (not (<= y.im 2.9e+60)))
   (/ x.re (- y.im))
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.5e+101) || !(y_46_im <= 2.9e+60)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-8.5d+101)) .or. (.not. (y_46im <= 2.9d+60))) then
        tmp = x_46re / -y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.5e+101) || !(y_46_im <= 2.9e+60)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.5e+101) or not (y_46_im <= 2.9e+60):
		tmp = x_46_re / -y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.5e+101) || !(y_46_im <= 2.9e+60))
		tmp = Float64(x_46_re / Float64(-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 <= -8.5e+101) || ~((y_46_im <= 2.9e+60)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -8.5e+101], N[Not[LessEqual[y$46$im, 2.9e+60]], $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 < -8.5000000000000001e101 or 2.9e60 < y.im

   1. Initial program 47.9%

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

   3. Taylor expanded in y.re around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

   5. Simplified 77.8%

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

   if -8.5000000000000001e101 < y.im < 2.9e60

   1. Initial program 76.0%

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

   3. Taylor expanded in y.re around inf 65.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+101} \lor \neg \left(y.im \leq 2.9 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.2% 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 44.9%

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

4. Final simplification 44.9%

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

Reproduce

herbie shell --seed 2024048 
(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))))