_divideComplex, imaginary part

Percentage Accurate: 61.2% → 85.6%

Time: 12.4s

Alternatives: 10

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: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y.re - y.im \cdot \frac{x.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3800000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (/ (* y.re (/ x.im (hypot y.im y.re))) (hypot y.im y.re))
          (/ x.re y.im))))
   (if (<= y.im -5e+56)
     t_0
     (if (<= y.im -1e-134)
       (*
        (/ (- y.re (* y.im (/ x.re x.im))) (hypot y.re y.im))
        (/ x.im (hypot y.re y.im)))
       (if (<= y.im 3800000.0)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -5e+56) {
		tmp = t_0;
	} else if (y_46_im <= -1e-134) {
		tmp = ((y_46_re - (y_46_im * (x_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 3800000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	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 / Math.hypot(y_46_im, y_46_re))) / Math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -5e+56) {
		tmp = t_0;
	} else if (y_46_im <= -1e-134) {
		tmp = ((y_46_re - (y_46_im * (x_46_re / x_46_im))) / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 3800000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * (x_46_im / math.hypot(y_46_im, y_46_re))) / math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -5e+56:
		tmp = t_0
	elif y_46_im <= -1e-134:
		tmp = ((y_46_re - (y_46_im * (x_46_re / x_46_im))) / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= 3800000.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -5e+56)
		tmp = t_0;
	elseif (y_46_im <= -1e-134)
		tmp = Float64(Float64(Float64(y_46_re - Float64(y_46_im * Float64(x_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 3800000.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5e+56)
		tmp = t_0;
	elseif (y_46_im <= -1e-134)
		tmp = ((y_46_re - (y_46_im * (x_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 3800000.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * N[(x$46$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] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5e+56], t$95$0, If[LessEqual[y$46$im, -1e-134], N[(N[(N[(y$46$re - N[(y$46$im * N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision], If[LessEqual[y$46$im, 3800000.0], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.00000000000000024e56 or 3.8e6 < y.im

   1. Initial program 47.8%

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

      1. div-sub 47.8%

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

      2. *-un-lft-identity 47.8%

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

      3. add-sqr-sqrt 47.8%

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

      4. times-frac 47.9%

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

      5. fma-neg 47.9%

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

      6. hypot-define 47.9%

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

      7. hypot-define 54.5%

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

      8. add-sqr-sqrt 54.5%

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

      9. pow2 54.5%

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

      10. hypot-define 54.5%

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

   4. Applied egg-rr 54.5%

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

      1. fma-neg 54.5%

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

   6. Simplified 71.2%

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

      1. associate-*l/ 71.3%

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

      2. *-un-lft-identity 71.3%

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

   8. Applied egg-rr 71.3%

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

   9. Taylor expanded in y.im around inf 88.1%

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

   if -5.00000000000000024e56 < y.im < -1.00000000000000004e-134

   1. Initial program 75.9%

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

   3. Taylor expanded in x.im around inf 71.4%

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

      1. mul-1-neg 71.4%

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

      2. unsub-neg 71.4%

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

      3. *-commutative 71.4%

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

      4. associate-/l* 69.1%

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

   5. Simplified 69.1%

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

      1. *-commutative 69.1%

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

      2. +-commutative 69.1%

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

      3. add-sqr-sqrt 69.1%

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

      4. hypot-undefine 69.1%

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

      5. hypot-undefine 69.1%

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

      6. times-frac 89.1%

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

      7. hypot-undefine 74.3%

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

      8. +-commutative 74.3%

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

      9. hypot-define 89.1%

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

      10. hypot-undefine 74.3%

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

      11. +-commutative 74.3%

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

      12. hypot-define 89.1%

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

   7. Applied egg-rr 89.1%

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

   if -1.00000000000000004e-134 < y.im < 3.8e6

   1. Initial program 66.2%

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

   3. Taylor expanded in y.re around inf 93.5%

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

      1. mul-1-neg 93.5%

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

      2. unsub-neg 93.5%

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

      3. associate-/l* 94.4%

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

   5. Simplified 94.4%

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

Alternative 2: 88.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+269}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, -y.im, x.im \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re - y.im \cdot \frac{x.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+269)
   (/
    (/ (fma x.re (- y.im) (* x.im y.re)) (hypot y.re y.im))
    (hypot y.re y.im))
   (*
    (/ (- y.re (* y.im (/ x.re x.im))) (hypot y.re y.im))
    (/ x.im (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+269) {
		tmp = (fma(x_46_re, -y_46_im, (x_46_im * y_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = ((y_46_re - (y_46_im * (x_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 5e+269)
		tmp = Float64(Float64(fma(x_46_re, Float64(-y_46_im), Float64(x_46_im * y_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(Float64(y_46_re - Float64(y_46_im * Float64(x_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 5e+269], N[(N[(N[(x$46$re * (-y$46$im) + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re - N[(y$46$im * N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000002e269

   1. Initial program 75.9%

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

      1. add-sqr-sqrt 75.9%

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

      2. associate-/r* 76.1%

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

      3. sub-neg 76.1%

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

      4. +-commutative 76.1%

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

      5. distribute-rgt-neg-in 76.1%

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

      6. fma-define 76.1%

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

      7. hypot-define 76.1%

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

      8. hypot-define 95.4%

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

   4. Applied egg-rr 95.4%

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

   if 5.0000000000000002e269 < (/.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 10.6%

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

   3. Taylor expanded in x.im around inf 10.6%

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

      1. mul-1-neg 10.6%

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

      2. unsub-neg 10.6%

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

      3. *-commutative 10.6%

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

      4. associate-/l* 10.6%

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

   5. Simplified 10.6%

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

      1. *-commutative 10.6%

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

      2. +-commutative 10.6%

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

      3. add-sqr-sqrt 10.6%

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

      4. hypot-undefine 10.6%

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

      5. hypot-undefine 10.6%

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

      6. times-frac 74.5%

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

      7. hypot-undefine 20.2%

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

      8. +-commutative 20.2%

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

      9. hypot-define 74.5%

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

      10. hypot-undefine 20.2%

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

      11. +-commutative 20.2%

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

      12. hypot-define 74.5%

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

   7. Applied egg-rr 74.5%

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

Alternative 3: 84.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -2.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 250000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (/ (* y.re (/ x.im (hypot y.im y.re))) (hypot y.im y.re))
          (/ x.re y.im))))
   (if (<= y.im -2.2e+96)
     t_0
     (if (<= y.im -2.25e-51)
       (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 250000000.0)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.2e+96) {
		tmp = t_0;
	} else if (y_46_im <= -2.25e-51) {
		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));
	} else if (y_46_im <= 250000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	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 / Math.hypot(y_46_im, y_46_re))) / Math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.2e+96) {
		tmp = t_0;
	} else if (y_46_im <= -2.25e-51) {
		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));
	} else if (y_46_im <= 250000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * (x_46_im / math.hypot(y_46_im, y_46_re))) / math.hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.2e+96:
		tmp = t_0
	elif y_46_im <= -2.25e-51:
		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))
	elif y_46_im <= 250000000.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.2e+96)
		tmp = t_0;
	elseif (y_46_im <= -2.25e-51)
		tmp = 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)));
	elseif (y_46_im <= 250000000.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * (x_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_im, y_46_re)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.2e+96)
		tmp = t_0;
	elseif (y_46_im <= -2.25e-51)
		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));
	elseif (y_46_im <= 250000000.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * N[(x$46$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] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+96], t$95$0, If[LessEqual[y$46$im, -2.25e-51], 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], If[LessEqual[y$46$im, 250000000.0], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.1999999999999999e96 or 2.5e8 < y.im

   1. Initial program 44.9%

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

      1. div-sub 44.9%

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

      2. *-un-lft-identity 44.9%

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

      3. add-sqr-sqrt 44.9%

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

      4. times-frac 44.9%

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

      5. fma-neg 44.9%

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

      6. hypot-define 44.9%

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

      7. hypot-define 52.2%

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

      8. add-sqr-sqrt 52.2%

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

      9. pow2 52.2%

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

      10. hypot-define 52.2%

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

   4. Applied egg-rr 52.2%

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

      1. fma-neg 52.2%

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

   6. Simplified 69.8%

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

      1. associate-*l/ 69.9%

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

      2. *-un-lft-identity 69.9%

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

   8. Applied egg-rr 69.9%

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

   9. Taylor expanded in y.im around inf 91.7%

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

   if -2.1999999999999999e96 < y.im < -2.24999999999999987e-51

   1. Initial program 83.0%

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

   if -2.24999999999999987e-51 < y.im < 2.5e8

   1. Initial program 65.6%

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

   3. Taylor expanded in y.re around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. unsub-neg 91.5%

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

      3. associate-/l* 92.3%

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

   5. Simplified 92.3%

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

Alternative 4: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1850:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))
   (if (<= y.im -1.7e+104)
     t_1
     (if (<= y.im -5.6e-52)
       t_0
       (if (<= y.im 1850.0)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 2.4e+79) 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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+104) {
		tmp = t_1;
	} else if (y_46_im <= -5.6e-52) {
		tmp = t_0;
	} else if (y_46_im <= 1850.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.4e+79) {
		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 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-1.7d+104)) then
        tmp = t_1
    else if (y_46im <= (-5.6d-52)) then
        tmp = t_0
    else if (y_46im <= 1850.0d0) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 2.4d+79) 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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.7e+104) {
		tmp = t_1;
	} else if (y_46_im <= -5.6e-52) {
		tmp = t_0;
	} else if (y_46_im <= 1850.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.4e+79) {
		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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.7e+104:
		tmp = t_1
	elif y_46_im <= -5.6e-52:
		tmp = t_0
	elif y_46_im <= 1850.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 2.4e+79:
		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(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)))
	t_1 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.7e+104)
		tmp = t_1;
	elseif (y_46_im <= -5.6e-52)
		tmp = t_0;
	elseif (y_46_im <= 1850.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2.4e+79)
		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 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.7e+104)
		tmp = t_1;
	elseif (y_46_im <= -5.6e-52)
		tmp = t_0;
	elseif (y_46_im <= 1850.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 2.4e+79)
		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[(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]}, Block[{t$95$1 = N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+104], t$95$1, If[LessEqual[y$46$im, -5.6e-52], t$95$0, If[LessEqual[y$46$im, 1850.0], 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$46$im, 2.4e+79], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.6999999999999998e104 or 2.39999999999999986e79 < y.im

   1. Initial program 36.0%

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

   3. Taylor expanded in y.re around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. unpow2 70.9%

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

      5. associate-/r* 75.2%

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

      6. div-sub 75.2%

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

      7. *-commutative 75.2%

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

   5. Simplified 75.2%

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

   6. Taylor expanded in y.im around inf 75.2%

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

      1. neg-mul-1 75.2%

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

      2. +-commutative 75.2%

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

      3. sub-neg 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-*r/ 82.8%

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

   8. Simplified 82.8%

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

   if -1.6999999999999998e104 < y.im < -5.59999999999999989e-52 or 1850 < y.im < 2.39999999999999986e79

   1. Initial program 86.2%

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

   if -5.59999999999999989e-52 < y.im < 1850

   1. Initial program 65.6%

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

   3. Taylor expanded in y.re around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. unsub-neg 91.5%

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

      3. associate-/l* 92.3%

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

   5. Simplified 92.3%

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

Alternative 5: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-51} \lor \neg \left(y.re \leq 7 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e-51) (not (<= y.re 7e-49)))
   (/ (- 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 <= -6.5e-51) || !(y_46_re <= 7e-49)) {
		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 <= (-6.5d-51)) .or. (.not. (y_46re <= 7d-49))) 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 <= -6.5e-51) || !(y_46_re <= 7e-49)) {
		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 <= -6.5e-51) or not (y_46_re <= 7e-49):
		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 <= -6.5e-51) || !(y_46_re <= 7e-49))
		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(Float64(x_46_im * 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 <= -6.5e-51) || ~((y_46_re <= 7e-49)))
		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, -6.5e-51], N[Not[LessEqual[y$46$re, 7e-49]], $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[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5000000000000003e-51 or 7.00000000000000012e-49 < y.re

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if -6.5000000000000003e-51 < y.re < 7.00000000000000012e-49

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 83.5%

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

      6. div-sub 83.6%

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

      7. *-commutative 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-51} \lor \neg \left(y.re \leq 7 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-51} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -7e-51) (not (<= y.re 3.5e-49)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -7e-51) || !(y_46_re <= 3.5e-49)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-7d-51)) .or. (.not. (y_46re <= 3.5d-49))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -7e-51) || !(y_46_re <= 3.5e-49)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -7e-51) or not (y_46_re <= 3.5e-49):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -7e-51) || !(y_46_re <= 3.5e-49))
		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(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -7e-51) || ~((y_46_re <= 3.5e-49)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -7e-51], N[Not[LessEqual[y$46$re, 3.5e-49]], $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[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.9999999999999995e-51 or 3.50000000000000006e-49 < y.re

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if -6.9999999999999995e-51 < y.re < 3.50000000000000006e-49

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 83.5%

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

      6. div-sub 83.6%

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

      7. *-commutative 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in y.im around inf 83.6%

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

      1. neg-mul-1 83.6%

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

      2. +-commutative 83.6%

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

      3. sub-neg 83.6%

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

      4. *-commutative 83.6%

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

      5. associate-*r/ 81.8%

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

   8. Simplified 81.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-51} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-51} \lor \neg \left(y.re \leq 1.3 \cdot 10^{-210}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.6e-51) (not (<= y.re 1.3e-210)))
   (/ (- x.im (* x.re (/ y.im y.re))) 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-51) || !(y_46_re <= 1.3e-210)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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-51)) .or. (.not. (y_46re <= 1.3d-210))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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-51) || !(y_46_re <= 1.3e-210)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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-51) or not (y_46_re <= 1.3e-210):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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-51) || !(y_46_re <= 1.3e-210))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(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 <= -1.6e-51) || ~((y_46_re <= 1.3e-210)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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-51], N[Not[LessEqual[y$46$re, 1.3e-210]], $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[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.6e-51 or 1.2999999999999999e-210 < y.re

   1. Initial program 55.7%

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

   3. Taylor expanded in y.re around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. unsub-neg 75.9%

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

      3. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

   if -1.6e-51 < y.re < 1.2999999999999999e-210

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. distribute-neg-frac2 81.9%

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

   5. Simplified 81.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-51} \lor \neg \left(y.re \leq 1.3 \cdot 10^{-210}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e-51)
   (/ (- x.im (* y.im (/ x.re y.re))) y.re)
   (if (<= y.re 4e-50)
     (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.6e-51) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4e-50) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-5.6d-51)) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else if (y_46re <= 4d-50) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.6e-51) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4e-50) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5.6e-51:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= 4e-50:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.6e-51)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= 4e-50)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5.6e-51)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= 4e-50)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.6e-51], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4e-50], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / 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 3 regimes

2. if y.re < -5.6e-51

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf 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. *-commutative 83.7%

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

      2. *-un-lft-identity 83.7%

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

      3. metadata-eval 83.7%

         \[\leadsto \frac{x.im + -1 \cdot \frac{y.im \cdot x.re}{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot y.re}}{y.re} \]

      4. times-frac 83.8%

         \[\leadsto \frac{x.im + -1 \cdot \color{blue}{\left(\frac{y.im}{3 \cdot 0.3333333333333333} \cdot \frac{x.re}{y.re}\right)}}{y.re} \]

      5. metadata-eval 83.8%

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

   5. Applied egg-rr 83.8%

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

   if -5.6e-51 < y.re < 4.00000000000000003e-50

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 83.5%

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

      6. div-sub 83.6%

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

      7. *-commutative 83.6%

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

   5. Simplified 83.6%

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

   if 4.00000000000000003e-50 < y.re

   1. Initial program 54.4%

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

   3. Taylor expanded in y.re around inf 77.3%

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

      1. mul-1-neg 77.3%

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

      2. unsub-neg 77.3%

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

      3. associate-/l* 82.5%

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

   5. Simplified 82.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-51} \lor \neg \left(y.re \leq 1.05 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.2e-51) (not (<= y.re 1.05e-46)))
   (/ 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 <= -6.2e-51) || !(y_46_re <= 1.05e-46)) {
		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 <= (-6.2d-51)) .or. (.not. (y_46re <= 1.05d-46))) 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 <= -6.2e-51) || !(y_46_re <= 1.05e-46)) {
		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 <= -6.2e-51) or not (y_46_re <= 1.05e-46):
		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 <= -6.2e-51) || !(y_46_re <= 1.05e-46))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(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 <= -6.2e-51) || ~((y_46_re <= 1.05e-46)))
		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, -6.2e-51], N[Not[LessEqual[y$46$re, 1.05e-46]], $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 < -6.1999999999999995e-51 or 1.04999999999999994e-46 < y.re

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around inf 67.8%

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

   if -6.1999999999999995e-51 < y.re < 1.04999999999999994e-46

   1. Initial program 68.5%

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

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

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

      1. mul-1-neg 73.9%

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

      2. distribute-neg-frac2 73.9%

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

   5. Simplified 73.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-51} \lor \neg \left(y.re \leq 1.05 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in y.re around inf 45.9%

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

Reproduce

herbie shell --seed 2024096 
(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))))