_divideComplex, imaginary part

Percentage Accurate: 62.5% → 88.6%

Time: 12.5s

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: 62.5% 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: 88.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) - \frac{y.im \cdot x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.2e+135)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 4.8e+159)
     (-
      (* (/ 1.0 (hypot y.re y.im)) (* x.im (/ y.re (hypot y.re y.im))))
      (/ (* y.im x.re) (pow (hypot y.re y.im) 2.0)))
     (/ (- (/ y.re (/ y.im x.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.2e+135) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 4.8e+159) {
		tmp = ((1.0 / hypot(y_46_re, y_46_im)) * (x_46_im * (y_46_re / hypot(y_46_re, y_46_im)))) - ((y_46_im * x_46_re) / pow(hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.2e+135) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 4.8e+159) {
		tmp = ((1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im * (y_46_re / Math.hypot(y_46_re, y_46_im)))) - ((y_46_im * x_46_re) / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.2e+135:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 4.8e+159:
		tmp = ((1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im * (y_46_re / math.hypot(y_46_re, y_46_im)))) - ((y_46_im * x_46_re) / math.pow(math.hypot(y_46_re, y_46_im), 2.0))
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.2e+135)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 4.8e+159)
		tmp = Float64(Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im * Float64(y_46_re / hypot(y_46_re, y_46_im)))) - Float64(Float64(y_46_im * x_46_re) / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_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_im <= -5.2e+135)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 4.8e+159)
		tmp = ((1.0 / hypot(y_46_re, y_46_im)) * (x_46_im * (y_46_re / hypot(y_46_re, y_46_im)))) - ((y_46_im * x_46_re) / (hypot(y_46_re, y_46_im) ^ 2.0));
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.2e+135], 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, 4.8e+159], N[(N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im * N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im * x$46$re), $MachinePrecision] / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.2e135

   1. Initial program 39.2%

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

   3. Taylor expanded in y.re around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. unpow2 85.7%

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

      5. associate-/r* 85.9%

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

      6. div-sub 85.9%

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

      7. *-commutative 85.9%

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

      8. associate-/l* 93.3%

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

   5. Simplified 93.3%

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

   if -5.2e135 < y.im < 4.8e159

   1. Initial program 71.4%

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

      1. div-sub 67.0%

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

      2. *-un-lft-identity 67.0%

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

      3. add-sqr-sqrt 67.0%

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

      4. times-frac 67.0%

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

      5. fmm-def 67.0%

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

      6. hypot-define 67.0%

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

      7. hypot-define 76.8%

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

      8. associate-/l* 76.4%

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

      9. add-sqr-sqrt 76.4%

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

      10. pow2 76.4%

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

      11. hypot-define 76.4%

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

   4. Applied egg-rr 76.4%

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

      1. fmm-undef 75.8%

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

      2. associate-/l* 91.8%

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

      3. associate-*r/ 92.2%

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

      4. *-commutative 92.2%

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

   6. Simplified 92.2%

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

   if 4.8e159 < y.im

   1. Initial program 35.9%

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

   3. Taylor expanded in y.re around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. mul-1-neg 67.8%

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

      3. unsub-neg 67.8%

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

      4. unpow2 67.8%

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

      5. associate-/r* 75.7%

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

      6. div-sub 75.7%

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

      7. *-commutative 75.7%

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

      8. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

      1. clear-num 87.0%

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

      2. un-div-inv 87.0%

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

   7. Applied egg-rr 87.0%

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

Alternative 2: 86.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{x.im + \frac{-1}{\frac{y.re}{y.im \cdot x.re}}}{y.re}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_1 (- INFINITY))
     (/ (+ x.im (/ -1.0 (/ y.re (* y.im x.re)))) y.re)
     (if (<= t_1 INFINITY)
       (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
       (/ (- (* 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 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} 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 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re
	elif t_1 <= math.inf:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	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(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_im + Float64(-1.0 / Float64(y_46_re / Float64(y_46_im * x_46_re)))) / y_46_re);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(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)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	elseif (t_1 <= Inf)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	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[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(x$46$im + N[(-1.0 / N[(y$46$re / N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(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 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 36.4%

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

      1. div-sub 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. add-sqr-sqrt 25.3%

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

      4. times-frac 25.3%

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

      5. fmm-def 25.3%

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

      6. hypot-define 25.3%

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

      7. hypot-define 25.3%

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

      8. associate-/l* 41.1%

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

      9. add-sqr-sqrt 41.1%

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

      10. pow2 41.1%

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

      11. hypot-define 41.1%

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

   4. Applied egg-rr 41.1%

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

      1. fmm-undef 41.1%

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

      2. associate-/l* 83.3%

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

      3. associate-*r/ 67.5%

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

      4. *-commutative 67.5%

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

   6. Simplified 67.5%

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

   7. Taylor expanded in y.re around inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. unsub-neg 84.1%

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

      3. associate-/l* 84.0%

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

   9. Simplified 84.0%

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

       1. associate-*r/ 84.1%

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

       2. clear-num 84.1%

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

       3. *-commutative 84.1%

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

   11. Applied egg-rr 84.1%

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

   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))) < +inf.0

   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
   3. Step-by-step derivation

      1. *-un-lft-identity 83.9%

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

      2. add-sqr-sqrt 83.8%

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

      3. times-frac 83.8%

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

      4. hypot-define 83.8%

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

      5. hypot-define 97.5%

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

   4. Applied egg-rr 97.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around 0 42.5%

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

      1. +-commutative 42.5%

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

      2. mul-1-neg 42.5%

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

      3. unsub-neg 42.5%

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

      4. unpow2 42.5%

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

      5. associate-/r* 42.9%

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

      6. div-sub 43.0%

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

      7. *-commutative 43.0%

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

      8. associate-/l* 56.7%

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

   5. Simplified 56.7%

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

4. Final simplification 87.3%

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

Alternative 3: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 800000000000:\\ \;\;\;\;\frac{x.im + \frac{-1}{\frac{y.re}{y.im \cdot x.re}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5e+101)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im -2.9e-71)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 800000000000.0)
       (/ (+ x.im (/ -1.0 (/ y.re (* y.im x.re)))) y.re)
       (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5e+101) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -2.9e-71) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 800000000000.0) {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_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_46im <= (-5d+101)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-2.9d-71)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 800000000000.0d0) then
        tmp = (x_46im + ((-1.0d0) / (y_46re / (y_46im * x_46re)))) / y_46re
    else
        tmp = ((y_46re / (y_46im / x_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_im <= -5e+101) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -2.9e-71) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 800000000000.0) {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5e+101:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -2.9e-71:
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 800000000000.0:
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5e+101)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -2.9e-71)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 800000000000.0)
		tmp = Float64(Float64(x_46_im + Float64(-1.0 / Float64(y_46_re / Float64(y_46_im * x_46_re)))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_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_im <= -5e+101)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -2.9e-71)
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 800000000000.0)
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5e+101], 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, -2.9e-71], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 800000000000.0], N[(N[(x$46$im + N[(-1.0 / N[(y$46$re / N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.99999999999999989e101

   1. Initial program 43.9%

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

   3. Taylor expanded in y.re around 0 83.7%

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

      1. +-commutative 83.7%

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

      2. mul-1-neg 83.7%

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

      3. unsub-neg 83.7%

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

      4. unpow2 83.7%

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

      5. associate-/r* 83.8%

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

      6. div-sub 83.8%

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

      7. *-commutative 83.8%

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

      8. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

   if -4.99999999999999989e101 < y.im < -2.8999999999999999e-71

   1. Initial program 77.6%

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

   if -2.8999999999999999e-71 < y.im < 8e11

   1. Initial program 72.5%

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

      1. div-sub 65.5%

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

      2. *-un-lft-identity 65.5%

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

      3. add-sqr-sqrt 65.5%

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

      4. times-frac 65.5%

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

      5. fmm-def 65.5%

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

      6. hypot-define 65.5%

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

      7. hypot-define 77.0%

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

      8. associate-/l* 74.5%

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

      9. add-sqr-sqrt 74.5%

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

      10. pow2 74.5%

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

      11. hypot-define 74.5%

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

   4. Applied egg-rr 74.5%

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

      1. fmm-undef 73.6%

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

      2. associate-/l* 88.8%

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

      3. associate-*r/ 91.3%

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

      4. *-commutative 91.3%

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

   6. Simplified 91.3%

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

   7. Taylor expanded in y.re around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. unsub-neg 92.2%

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

      3. associate-/l* 90.8%

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

   9. Simplified 90.8%

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

       1. associate-*r/ 92.2%

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

       2. clear-num 92.2%

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

       3. *-commutative 92.2%

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

   11. Applied egg-rr 92.2%

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

   if 8e11 < y.im

   1. Initial program 45.0%

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

   3. Taylor expanded in y.re around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. mul-1-neg 63.6%

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

      3. unsub-neg 63.6%

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

      4. unpow2 63.6%

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

      5. associate-/r* 68.2%

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

      6. div-sub 68.2%

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

      7. *-commutative 68.2%

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

      8. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. clear-num 74.9%

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

      2. un-div-inv 74.9%

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

   7. Applied egg-rr 74.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 800000000000:\\ \;\;\;\;\frac{x.im + \frac{-1}{\frac{y.re}{y.im \cdot x.re}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-26} \lor \neg \left(y.im \leq 105000000\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{-1}{\frac{y.re}{y.im \cdot x.re}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.5e-26) (not (<= y.im 105000000.0)))
   (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)
   (/ (+ x.im (/ -1.0 (/ y.re (* y.im x.re)))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e-26) || !(y_46_im <= 105000000.0)) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.5d-26)) .or. (.not. (y_46im <= 105000000.0d0))) then
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im + ((-1.0d0) / (y_46re / (y_46im * x_46re)))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e-26) || !(y_46_im <= 105000000.0)) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.5e-26) or not (y_46_im <= 105000000.0):
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.5e-26) || !(y_46_im <= 105000000.0))
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im + Float64(-1.0 / Float64(y_46_re / Float64(y_46_im * x_46_re)))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.5e-26) || ~((y_46_im <= 105000000.0)))
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im + (-1.0 / (y_46_re / (y_46_im * x_46_re)))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.5e-26], N[Not[LessEqual[y$46$im, 105000000.0]], $MachinePrecision]], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im + N[(-1.0 / N[(y$46$re / N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5000000000000001e-26 or 1.05e8 < y.im

   1. Initial program 51.7%

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

   3. Taylor expanded in y.re around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

      4. unpow2 69.3%

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

      5. associate-/r* 71.4%

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

      6. div-sub 71.4%

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

      7. *-commutative 71.4%

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

      8. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

   7. Applied egg-rr 76.5%

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

   if -2.5000000000000001e-26 < y.im < 1.05e8

   1. Initial program 72.9%

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

      1. div-sub 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. add-sqr-sqrt 66.1%

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

      4. times-frac 66.1%

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

      5. fmm-def 66.1%

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

      6. hypot-define 66.1%

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

      7. hypot-define 77.2%

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

      8. associate-/l* 74.8%

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

      9. add-sqr-sqrt 74.8%

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

      10. pow2 74.8%

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

      11. hypot-define 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. fmm-undef 73.9%

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

      2. associate-/l* 89.3%

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

      3. associate-*r/ 91.7%

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

      4. *-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y.re around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. associate-/l* 89.6%

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

   9. Simplified 89.6%

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

       1. associate-*r/ 91.0%

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

       2. clear-num 91.0%

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

       3. *-commutative 91.0%

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

   11. Applied egg-rr 91.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-26} \lor \neg \left(y.im \leq 105000000\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{-1}{\frac{y.re}{y.im \cdot x.re}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{-25} \lor \neg \left(y.im \leq 1500000000000\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2e-25) (not (<= y.im 1500000000000.0)))
   (/ (- (/ y.re (/ y.im x.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_im <= -2e-25) || !(y_46_im <= 1500000000000.0)) {
		tmp = ((y_46_re / (y_46_im / x_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_46im <= (-2d-25)) .or. (.not. (y_46im <= 1500000000000.0d0))) then
        tmp = ((y_46re / (y_46im / x_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_im <= -2e-25) || !(y_46_im <= 1500000000000.0)) {
		tmp = ((y_46_re / (y_46_im / x_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_im <= -2e-25) or not (y_46_im <= 1500000000000.0):
		tmp = ((y_46_re / (y_46_im / x_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_im <= -2e-25) || !(y_46_im <= 1500000000000.0))
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_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_im <= -2e-25) || ~((y_46_im <= 1500000000000.0)))
		tmp = ((y_46_re / (y_46_im / x_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[Or[LessEqual[y$46$im, -2e-25], N[Not[LessEqual[y$46$im, 1500000000000.0]], $MachinePrecision]], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $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 2 regimes

2. if y.im < -2.00000000000000008e-25 or 1.5e12 < y.im

   1. Initial program 51.7%

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

   3. Taylor expanded in y.re around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

      4. unpow2 69.3%

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

      5. associate-/r* 71.4%

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

      6. div-sub 71.4%

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

      7. *-commutative 71.4%

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

      8. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

   7. Applied egg-rr 76.5%

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

   if -2.00000000000000008e-25 < y.im < 1.5e12

   1. Initial program 72.9%

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

      1. div-sub 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. add-sqr-sqrt 66.1%

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

      4. times-frac 66.1%

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

      5. fmm-def 66.1%

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

      6. hypot-define 66.1%

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

      7. hypot-define 77.2%

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

      8. associate-/l* 74.8%

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

      9. add-sqr-sqrt 74.8%

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

      10. pow2 74.8%

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

      11. hypot-define 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. fmm-undef 73.9%

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

      2. associate-/l* 89.3%

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

      3. associate-*r/ 91.7%

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

      4. *-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y.re around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. associate-/l* 89.6%

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

   9. Simplified 89.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{-25} \lor \neg \left(y.im \leq 1500000000000\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-25} \lor \neg \left(y.im \leq 2700000000\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3e-25) (not (<= y.im 2700000000.0)))
   (/ (- (* y.re (/ x.im 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_im <= -3e-25) || !(y_46_im <= 2700000000.0)) {
		tmp = ((y_46_re * (x_46_im / 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_46im <= (-3d-25)) .or. (.not. (y_46im <= 2700000000.0d0))) then
        tmp = ((y_46re * (x_46im / 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_im <= -3e-25) || !(y_46_im <= 2700000000.0)) {
		tmp = ((y_46_re * (x_46_im / 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_im <= -3e-25) or not (y_46_im <= 2700000000.0):
		tmp = ((y_46_re * (x_46_im / 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_im <= -3e-25) || !(y_46_im <= 2700000000.0))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - 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 <= -3e-25) || ~((y_46_im <= 2700000000.0)))
		tmp = ((y_46_re * (x_46_im / 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[Or[LessEqual[y$46$im, -3e-25], N[Not[LessEqual[y$46$im, 2700000000.0]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - 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 < -2.9999999999999998e-25 or 2.7e9 < y.im

   1. Initial program 51.7%

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

   3. Taylor expanded in y.re around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

      4. unpow2 69.3%

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

      5. associate-/r* 71.4%

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

      6. div-sub 71.4%

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

      7. *-commutative 71.4%

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

      8. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

   if -2.9999999999999998e-25 < y.im < 2.7e9

   1. Initial program 72.9%

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

      1. div-sub 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. add-sqr-sqrt 66.1%

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

      4. times-frac 66.1%

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

      5. fmm-def 66.1%

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

      6. hypot-define 66.1%

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

      7. hypot-define 77.2%

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

      8. associate-/l* 74.8%

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

      9. add-sqr-sqrt 74.8%

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

      10. pow2 74.8%

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

      11. hypot-define 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. fmm-undef 73.9%

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

      2. associate-/l* 89.3%

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

      3. associate-*r/ 91.7%

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

      4. *-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y.re around inf 91.0%

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

      1. mul-1-neg 91.0%

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

      2. unsub-neg 91.0%

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

      3. associate-/l* 89.6%

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

   9. Simplified 89.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-25} \lor \neg \left(y.im \leq 2700000000\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+63} \lor \neg \left(y.im \leq 1.95 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.6e+63) (not (<= y.im 1.95e+167)))
   (/ (- 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 <= -3.6e+63) || !(y_46_im <= 1.95e+167)) {
		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 <= (-3.6d+63)) .or. (.not. (y_46im <= 1.95d+167))) 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 <= -3.6e+63) || !(y_46_im <= 1.95e+167)) {
		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 <= -3.6e+63) or not (y_46_im <= 1.95e+167):
		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 <= -3.6e+63) || !(y_46_im <= 1.95e+167))
		tmp = Float64(Float64(-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_im <= -3.6e+63) || ~((y_46_im <= 1.95e+167)))
		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, -3.6e+63], N[Not[LessEqual[y$46$im, 1.95e+167]], $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 < -3.59999999999999999e63 or 1.9499999999999999e167 < y.im

   1. Initial program 43.4%

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

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

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   5. Simplified 81.9%

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

   if -3.59999999999999999e63 < y.im < 1.9499999999999999e167

   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. Step-by-step derivation

      1. div-sub 65.3%

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

      2. *-un-lft-identity 65.3%

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

      3. add-sqr-sqrt 65.3%

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

      4. times-frac 65.3%

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

      5. fmm-def 65.3%

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

      6. hypot-define 65.3%

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

      7. hypot-define 75.5%

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

      8. associate-/l* 75.1%

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

      9. add-sqr-sqrt 75.1%

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

      10. pow2 75.1%

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

      11. hypot-define 75.1%

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

   4. Applied egg-rr 75.1%

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

      1. fmm-undef 74.5%

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

      2. associate-/l* 89.9%

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

      3. associate-*r/ 90.3%

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

      4. *-commutative 90.3%

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

   6. Simplified 90.3%

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

   7. Taylor expanded in y.re around inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. unsub-neg 75.2%

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

      3. associate-/l* 76.5%

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

   9. Simplified 76.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+63} \lor \neg \left(y.im \leq 1.95 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+74} \lor \neg \left(y.re \leq 200000\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.8e+74) (not (<= y.re 200000.0)))
   (/ 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 <= -6.8e+74) || !(y_46_re <= 200000.0)) {
		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 <= (-6.8d+74)) .or. (.not. (y_46re <= 200000.0d0))) 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 <= -6.8e+74) || !(y_46_re <= 200000.0)) {
		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 <= -6.8e+74) or not (y_46_re <= 200000.0):
		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 <= -6.8e+74) || !(y_46_re <= 200000.0))
		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 <= -6.8e+74) || ~((y_46_re <= 200000.0)))
		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, -6.8e+74], N[Not[LessEqual[y$46$re, 200000.0]], $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 < -6.7999999999999998e74 or 2e5 < y.re

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around inf 77.0%

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

   if -6.7999999999999998e74 < y.re < 2e5

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around 0 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

   5. Simplified 65.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+74} \lor \neg \left(y.re \leq 200000\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+152} \lor \neg \left(y.im \leq 2.15 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.7e+152) (not (<= y.im 2.15e+206)))
   (/ x.re y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.7e+152) || !(y_46_im <= 2.15e+206)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.7d+152)) .or. (.not. (y_46im <= 2.15d+206))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.7e+152) || !(y_46_im <= 2.15e+206)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.7e+152) or not (y_46_im <= 2.15e+206):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.7e+152) || !(y_46_im <= 2.15e+206))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.7e+152) || ~((y_46_im <= 2.15e+206)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.7e+152], N[Not[LessEqual[y$46$im, 2.15e+206]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.7000000000000001e152 or 2.14999999999999994e206 < y.im

   1. Initial program 41.1%

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

   3. Taylor expanded in y.re around 0 83.1%

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

      1. +-commutative 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. unpow2 83.1%

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

      5. associate-/r* 85.8%

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

      6. div-sub 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-/l* 97.2%

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

   5. Simplified 97.2%

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

      1. clear-num 94.7%

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

      2. sub-neg 94.7%

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

      3. associate-*r/ 83.4%

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

      4. *-commutative 83.4%

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

      5. mul-1-neg 83.4%

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

      6. +-commutative 83.4%

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

      7. inv-pow 83.4%

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

      8. +-commutative 83.4%

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

      9. add-sqr-sqrt 54.0%

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

      10. sqrt-unprod 50.3%

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

      11. mul-1-neg 50.3%

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

      12. mul-1-neg 50.3%

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

      13. sqr-neg 50.3%

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

      14. sqrt-unprod 12.0%

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

      15. add-sqr-sqrt 42.2%

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

      16. *-commutative 42.2%

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

      17. associate-*r/ 48.7%

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

      18. fma-define 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. unpow-1 48.7%

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

      2. fma-define 48.7%

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

      3. *-commutative 48.7%

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

      4. associate-*l/ 42.2%

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

      5. associate-*r/ 48.6%

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

      6. fma-undefine 48.6%

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

   9. Simplified 48.6%

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

   10. Taylor expanded in y.im around inf 39.6%

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

   if -1.7000000000000001e152 < y.im < 2.14999999999999994e206

   1. Initial program 67.9%

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

   3. Taylor expanded in y.re around inf 55.4%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+152} \lor \neg \left(y.im \leq 2.15 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.3% 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 61.5%

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

3. Taylor expanded in y.re around inf 45.5%

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

Reproduce

herbie shell --seed 2024172 
(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))))