_divideComplex, imaginary part

Percentage Accurate: 62.4% → 96.0%

Time: 10.0s

Alternatives: 9

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% 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 64.7%

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

   1. div-sub 62.3%

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

   2. *-commutative 62.3%

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

   3. add-sqr-sqrt 62.3%

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

   4. times-frac 64.8%

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

   5. fma-neg 64.8%

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

   6. hypot-define 64.8%

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

   7. hypot-define 81.9%

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

   8. associate-/l* 84.4%

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

   9. add-sqr-sqrt 84.4%

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

   10. pow2 84.4%

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

   11. hypot-define 84.4%

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

4. Applied egg-rr 84.4%

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

   1. *-un-lft-identity 84.4%

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

   2. unpow2 84.4%

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

   3. times-frac 95.5%

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

   4. hypot-undefine 84.4%

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

   5. +-commutative 84.4%

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

   6. hypot-undefine 95.5%

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

   7. hypot-undefine 84.4%

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

   8. +-commutative 84.4%

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

   9. hypot-undefine 95.5%

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

6. Applied egg-rr 95.5%

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

   1. associate-*l/ 95.5%

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

   2. *-un-lft-identity 95.5%

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

8. Applied egg-rr 95.5%

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

9. Final simplification 95.5%

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

Alternative 2: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (* (/ y.im (pow (hypot y.re y.im) 2.0)) (- x.re)))
     (if (<= t_0 2e+258)
       (*
        (/ 1.0 (hypot y.re y.im))
        (/ (fma x.im y.re (* x.re (- y.im))) (hypot y.re y.im)))
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((y_46_im / pow(hypot(y_46_re, y_46_im), 2.0)) * -x_46_re));
	} else if (t_0 <= 2e+258) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(y_46_im / (hypot(y_46_re, y_46_im) ^ 2.0)) * Float64(-x_46_re)));
	elseif (t_0 <= 2e+258)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], 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[(y$46$im / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+258], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 33.6%

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

      1. div-sub 13.6%

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

      2. *-commutative 13.6%

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

      3. add-sqr-sqrt 13.6%

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

      4. times-frac 51.7%

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

      5. fma-neg 51.7%

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

      6. hypot-define 51.7%

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

      7. hypot-define 71.7%

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

      8. associate-/l* 99.8%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow2 99.8%

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

      11. hypot-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.00000000000000011e258

   1. Initial program 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. add-sqr-sqrt 80.6%

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

      3. times-frac 80.6%

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

      4. hypot-define 80.6%

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

      5. fma-neg 80.6%

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

      6. distribute-rgt-neg-in 80.6%

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

      7. hypot-define 98.6%

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

   4. Applied egg-rr 98.6%

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

   if 2.00000000000000011e258 < (/.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 19.3%

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

   3. Taylor expanded in y.re around inf 65.4%

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

      1. mul-1-neg 65.4%

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

      2. unsub-neg 65.4%

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

      3. unsub-neg 65.4%

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

      4. remove-double-neg 65.4%

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

      5. mul-1-neg 65.4%

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

      6. neg-mul-1 65.4%

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

      7. mul-1-neg 65.4%

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

      8. distribute-lft-in 65.4%

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

      9. distribute-lft-in 65.4%

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

      10. mul-1-neg 65.4%

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

      11. unsub-neg 65.4%

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

      12. neg-mul-1 65.4%

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

      13. mul-1-neg 65.4%

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

      14. remove-double-neg 65.4%

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

      15. associate-/l* 73.5%

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

   5. Simplified 73.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
      2e+258)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.im y.re (* x.re (- y.im))) (hypot y.re y.im)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+258) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_im, y_46_re, (x_46_re * -y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 2e+258)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_im, y_46_re, Float64(x_46_re * Float64(-y_46_im))) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 2e+258], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im * y$46$re + N[(x$46$re * (-y$46$im)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 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))) < 2.00000000000000011e258

   1. Initial program 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. add-sqr-sqrt 78.2%

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

      3. times-frac 78.2%

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

      4. hypot-define 78.2%

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

      5. fma-neg 78.2%

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

      6. distribute-rgt-neg-in 78.2%

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

      7. hypot-define 95.3%

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

   4. Applied egg-rr 95.3%

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

   if 2.00000000000000011e258 < (/.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 19.3%

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

   3. Taylor expanded in y.re around inf 65.4%

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

      1. mul-1-neg 65.4%

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

      2. unsub-neg 65.4%

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

      3. unsub-neg 65.4%

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

      4. remove-double-neg 65.4%

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

      5. mul-1-neg 65.4%

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

      6. neg-mul-1 65.4%

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

      7. mul-1-neg 65.4%

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

      8. distribute-lft-in 65.4%

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

      9. distribute-lft-in 65.4%

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

      10. mul-1-neg 65.4%

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

      11. unsub-neg 65.4%

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

      12. neg-mul-1 65.4%

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

      13. mul-1-neg 65.4%

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

      14. remove-double-neg 65.4%

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

      15. associate-/l* 73.5%

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

   5. Simplified 73.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.im, y.re, x.re \cdot \left(-y.im\right)\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -8.5e+56)
     t_1
     (if (<= y.re -6e-95)
       t_0
       (if (<= y.re 1.7e-128)
         (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
         (if (<= y.re 1.65e+99) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8.5e+56) {
		tmp = t_1;
	} else if (y_46_re <= -6e-95) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e-128) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.65e+99) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-8.5d+56)) then
        tmp = t_1
    else if (y_46re <= (-6d-95)) then
        tmp = t_0
    else if (y_46re <= 1.7d-128) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) / y_46im
    else if (y_46re <= 1.65d+99) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8.5e+56) {
		tmp = t_1;
	} else if (y_46_re <= -6e-95) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e-128) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.65e+99) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -8.5e+56:
		tmp = t_1
	elif y_46_re <= -6e-95:
		tmp = t_0
	elif y_46_re <= 1.7e-128:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_re <= 1.65e+99:
		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(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -8.5e+56)
		tmp = t_1;
	elseif (y_46_re <= -6e-95)
		tmp = t_0;
	elseif (y_46_re <= 1.7e-128)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.65e+99)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -8.5e+56)
		tmp = t_1;
	elseif (y_46_re <= -6e-95)
		tmp = t_0;
	elseif (y_46_re <= 1.7e-128)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.65e+99)
		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[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -8.5e+56], t$95$1, If[LessEqual[y$46$re, -6e-95], t$95$0, If[LessEqual[y$46$re, 1.7e-128], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+99], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.4999999999999998e56 or 1.65e99 < y.re

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. unsub-neg 83.7%

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

      4. remove-double-neg 83.7%

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

      5. mul-1-neg 83.7%

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

      6. neg-mul-1 83.7%

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

      7. mul-1-neg 83.7%

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

      8. distribute-lft-in 83.7%

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

      9. distribute-lft-in 83.7%

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

      10. mul-1-neg 83.7%

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

      11. unsub-neg 83.7%

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

      12. neg-mul-1 83.7%

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

      13. mul-1-neg 83.7%

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

      14. remove-double-neg 83.7%

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

      15. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

   if -8.4999999999999998e56 < y.re < -6e-95 or 1.69999999999999987e-128 < y.re < 1.65e99

   1. Initial program 83.9%

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

   if -6e-95 < y.re < 1.69999999999999987e-128

   1. Initial program 76.1%

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

   3. Taylor expanded in y.re around 0 86.2%

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

      1. +-commutative 86.2%

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

      2. mul-1-neg 86.2%

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

      3. unsub-neg 86.2%

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

      4. unpow2 86.2%

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

      5. associate-/r* 93.5%

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

      6. div-sub 93.5%

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

      7. associate-/l* 95.5%

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

   5. Simplified 95.5%

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

      1. clear-num 95.5%

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

      2. un-div-inv 95.6%

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-95}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+99}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-93} \lor \neg \left(y.re \leq 1.3 \cdot 10^{+18} \lor \neg \left(y.re \leq 3.9 \cdot 10^{+65}\right) \land y.re \leq 1.6 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.3e-93)
         (not
          (or (<= y.re 1.3e+18)
              (and (not (<= y.re 3.9e+65)) (<= y.re 1.6e+79)))))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.3e-93) || !((y_46_re <= 1.3e+18) || (!(y_46_re <= 3.9e+65) && (y_46_re <= 1.6e+79)))) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.3d-93)) .or. (.not. (y_46re <= 1.3d+18) .or. (.not. (y_46re <= 3.9d+65)) .and. (y_46re <= 1.6d+79))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.3e-93) || !((y_46_re <= 1.3e+18) || (!(y_46_re <= 3.9e+65) && (y_46_re <= 1.6e+79)))) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.3e-93) or not ((y_46_re <= 1.3e+18) or (not (y_46_re <= 3.9e+65) and (y_46_re <= 1.6e+79))):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.3e-93) || !((y_46_re <= 1.3e+18) || (!(y_46_re <= 3.9e+65) && (y_46_re <= 1.6e+79))))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.3e-93) || ~(((y_46_re <= 1.3e+18) || (~((y_46_re <= 3.9e+65)) && (y_46_re <= 1.6e+79)))))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / 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, -1.3e-93], N[Not[Or[LessEqual[y$46$re, 1.3e+18], And[N[Not[LessEqual[y$46$re, 3.9e+65]], $MachinePrecision], LessEqual[y$46$re, 1.6e+79]]]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.2999999999999999e-93 or 1.3e18 < y.re < 3.8999999999999998e65 or 1.60000000000000001e79 < y.re

   1. Initial program 54.7%

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

   3. Taylor expanded in y.re around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. remove-double-neg 80.2%

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

      5. mul-1-neg 80.2%

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

      6. neg-mul-1 80.2%

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

      7. mul-1-neg 80.2%

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

      8. distribute-lft-in 80.2%

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

      9. distribute-lft-in 80.2%

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

      10. mul-1-neg 80.2%

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

      11. unsub-neg 80.2%

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

      12. neg-mul-1 80.2%

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

      13. mul-1-neg 80.2%

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

      14. remove-double-neg 80.2%

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

      15. associate-/l* 81.8%

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

   5. Simplified 81.8%

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

   if -1.2999999999999999e-93 < y.re < 1.3e18 or 3.8999999999999998e65 < y.re < 1.60000000000000001e79

   1. Initial program 78.1%

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

   3. Taylor expanded in y.re around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

      4. unpow2 77.7%

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

      5. associate-/r* 85.2%

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

      6. div-sub 85.2%

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

      7. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-93} \lor \neg \left(y.re \leq 1.3 \cdot 10^{+18} \lor \neg \left(y.re \leq 3.9 \cdot 10^{+65}\right) \land y.re \leq 1.6 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+66} \lor \neg \left(y.re \leq 1.02 \cdot 10^{+80}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -1.6e-93)
     t_0
     (if (<= y.re 4.4e+21)
       (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
       (if (or (<= y.re 1.8e+66) (not (<= y.re 1.02e+80)))
         t_0
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.6e-93) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e+21) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if ((y_46_re <= 1.8e+66) || !(y_46_re <= 1.02e+80)) {
		tmp = t_0;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-1.6d-93)) then
        tmp = t_0
    else if (y_46re <= 4.4d+21) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if ((y_46re <= 1.8d+66) .or. (.not. (y_46re <= 1.02d+80))) then
        tmp = t_0
    else
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -1.6e-93) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e+21) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if ((y_46_re <= 1.8e+66) || !(y_46_re <= 1.02e+80)) {
		tmp = t_0;
	} else {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -1.6e-93:
		tmp = t_0
	elif y_46_re <= 4.4e+21:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif (y_46_re <= 1.8e+66) or not (y_46_re <= 1.02e+80):
		tmp = t_0
	else:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.6e-93)
		tmp = t_0;
	elseif (y_46_re <= 4.4e+21)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif ((y_46_re <= 1.8e+66) || !(y_46_re <= 1.02e+80))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.6e-93)
		tmp = t_0;
	elseif (y_46_re <= 4.4e+21)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif ((y_46_re <= 1.8e+66) || ~((y_46_re <= 1.02e+80)))
		tmp = t_0;
	else
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e-93], t$95$0, If[LessEqual[y$46$re, 4.4e+21], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[Or[LessEqual[y$46$re, 1.8e+66], N[Not[LessEqual[y$46$re, 1.02e+80]], $MachinePrecision]], t$95$0, N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.5999999999999999e-93 or 4.4e21 < y.re < 1.8e66 or 1.02e80 < y.re

   1. Initial program 54.7%

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

   3. Taylor expanded in y.re around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. remove-double-neg 80.2%

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

      5. mul-1-neg 80.2%

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

      6. neg-mul-1 80.2%

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

      7. mul-1-neg 80.2%

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

      8. distribute-lft-in 80.2%

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

      9. distribute-lft-in 80.2%

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

      10. mul-1-neg 80.2%

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

      11. unsub-neg 80.2%

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

      12. neg-mul-1 80.2%

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

      13. mul-1-neg 80.2%

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

      14. remove-double-neg 80.2%

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

      15. associate-/l* 81.8%

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

   5. Simplified 81.8%

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

   if -1.5999999999999999e-93 < y.re < 4.4e21

   1. Initial program 77.9%

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

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

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. unpow2 76.6%

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

      5. associate-/r* 84.4%

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

      6. div-sub 84.5%

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

      7. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

   if 1.8e66 < y.re < 1.02e80

   1. Initial program 82.1%

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

      1. div-sub 82.1%

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

      2. *-commutative 82.1%

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

      3. add-sqr-sqrt 82.1%

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

      4. times-frac 82.1%

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

      5. fma-neg 82.1%

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

      6. hypot-define 82.1%

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

      7. hypot-define 82.1%

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

      8. associate-/l* 82.1%

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

      9. add-sqr-sqrt 82.1%

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

      10. pow2 82.1%

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

      11. hypot-define 82.1%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in y.im around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+66} \lor \neg \left(y.re \leq 1.02 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+89} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.5e+89) (not (<= y.im 4.6e+113)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e+89) || !(y_46_im <= 4.6e+113)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.5d+89)) .or. (.not. (y_46im <= 4.6d+113))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e+89) || !(y_46_im <= 4.6e+113)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.5e+89) or not (y_46_im <= 4.6e+113):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.5e+89) || !(y_46_im <= 4.6e+113))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.5e+89) || ~((y_46_im <= 4.6e+113)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.5e+89], N[Not[LessEqual[y$46$im, 4.6e+113]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.49999999999999992e89 or 4.59999999999999993e113 < y.im

   1. Initial program 48.4%

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

   3. Taylor expanded in y.re around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   5. Simplified 77.4%

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

   if -2.49999999999999992e89 < y.im < 4.59999999999999993e113

   1. Initial program 70.5%

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

   3. Taylor expanded in y.re around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. unsub-neg 74.1%

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

      4. remove-double-neg 74.1%

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

      5. mul-1-neg 74.1%

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

      6. neg-mul-1 74.1%

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

      7. mul-1-neg 74.1%

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

      8. distribute-lft-in 74.1%

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

      9. distribute-lft-in 74.1%

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

      10. mul-1-neg 74.1%

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

      11. unsub-neg 74.1%

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

      12. neg-mul-1 74.1%

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

      13. mul-1-neg 74.1%

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

      14. remove-double-neg 74.1%

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

      15. associate-/l* 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+89} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-97} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.6e-97) (not (<= y.re 1.15e-51)))
   (/ x.im y.re)
   (/ x.re (- y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.6e-97) || !(y_46_re <= 1.15e-51)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.6d-97)) .or. (.not. (y_46re <= 1.15d-51))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.6e-97) || !(y_46_re <= 1.15e-51)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.6e-97) or not (y_46_re <= 1.15e-51):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.6e-97) || !(y_46_re <= 1.15e-51))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.6e-97) || ~((y_46_re <= 1.15e-51)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.6e-97], N[Not[LessEqual[y$46$re, 1.15e-51]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.5999999999999999e-97 or 1.15000000000000001e-51 < y.re

   1. Initial program 58.8%

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

   3. Taylor expanded in y.re around inf 65.1%

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

   if -1.5999999999999999e-97 < y.re < 1.15000000000000001e-51

   1. Initial program 76.4%

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

   3. Taylor expanded in y.re around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   5. Simplified 68.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-97} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.1% 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 64.7%

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

3. Taylor expanded in y.re around inf 46.8%

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

Reproduce

herbie shell --seed 2024087 
(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))))