_divideComplex, imaginary part

Percentage Accurate: 62.2% → 89.5%

Time: 16.2s

Alternatives: 15

Speedup: 0.7×

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.9 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.9e+156)
   (* (/ 1.0 (hypot y.re y.im)) (- x.re (* (/ y.re y.im) x.im)))
   (if (<= y.im 1.95e+142)
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (* x.re (/ y.im (- (pow (hypot y.re y.im) 2.0)))))
     (- (/ (/ y.re y.im) (/ 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 <= -5.9e+156) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	} else if (y_46_im <= 1.95e+142) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (x_46_re * (y_46_im / -pow(hypot(y_46_re, y_46_im), 2.0))));
	} else {
		tmp = ((y_46_re / y_46_im) / (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 <= -5.9e+156)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - Float64(Float64(y_46_re / y_46_im) * x_46_im)));
	elseif (y_46_im <= 1.95e+142)
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re * Float64(y_46_im / Float64(-(hypot(y_46_re, y_46_im) ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.9e+156], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.95e+142], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(y$46$im / (-N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.8999999999999997e156

   1. Initial program 26.2%

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

      1. *-un-lft-identity 26.2%

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

      2. add-sqr-sqrt 26.2%

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

      3. times-frac 26.2%

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

      4. hypot-define 26.2%

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

      5. fma-neg 26.2%

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

      6. distribute-rgt-neg-in 26.2%

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

      7. hypot-define 47.9%

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

   4. Applied egg-rr 47.9%

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

   5. Taylor expanded in y.im around -inf 82.8%

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

      1. associate-/l* 87.1%

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

      2. associate-*r* 87.1%

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

      3. neg-mul-1 87.1%

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

      4. *-commutative 87.1%

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

   7. Simplified 87.1%

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

   if -5.8999999999999997e156 < y.im < 1.95e142

   1. Initial program 70.7%

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

      1. div-sub 67.9%

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

      2. *-commutative 67.9%

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

      3. add-sqr-sqrt 67.9%

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

      4. times-frac 71.6%

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

      5. fma-neg 71.6%

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

      6. hypot-define 71.6%

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

      7. hypot-define 89.4%

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

      8. associate-/l* 91.9%

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

      9. add-sqr-sqrt 91.9%

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

      10. pow2 91.9%

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

      11. hypot-define 91.9%

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

   4. Applied egg-rr 91.9%

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

   if 1.95e142 < y.im

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around 0 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. pow2 77.3%

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

      3. times-frac 88.5%

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

   7. Applied egg-rr 88.5%

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

      1. associate-*r* 94.7%

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

      2. clear-num 94.7%

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

      3. un-div-inv 94.7%

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

      4. un-div-inv 94.8%

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

   9. Applied egg-rr 94.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.9 \cdot 10^{+156}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, x.re \cdot \frac{y.im}{-{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. add-sqr-sqrt 77.1%

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

      3. times-frac 77.1%

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

      4. hypot-define 77.1%

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

      5. fma-neg 77.1%

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

      6. distribute-rgt-neg-in 77.1%

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

      7. hypot-define 94.0%

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

   4. Applied egg-rr 94.0%

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

   if +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 x.im around inf 1.5%

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

      1. *-commutative 1.5%

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

   5. Simplified 1.5%

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

      1. add-sqr-sqrt 1.5%

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

      2. hypot-undefine 1.5%

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

      3. hypot-undefine 1.5%

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

      4. frac-times 59.6%

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

      5. hypot-undefine 4.2%

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

      6. +-commutative 4.2%

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

      7. hypot-define 59.6%

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

      8. hypot-undefine 4.2%

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

      9. +-commutative 4.2%

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

      10. hypot-define 59.6%

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

   7. Applied egg-rr 59.6%

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

4. Final simplification 87.0%

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

Alternative 3: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.im \cdot x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{-127}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.05e+76)
   (* (/ 1.0 (hypot y.re y.im)) (- (/ (* y.im x.re) y.re) x.im))
   (if (<= y.re -1.85e-127)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 4.2e+31)
       (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im))
       (* (/ y.re (hypot y.im y.re)) (/ x.im (hypot y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.05e+76) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	} else if (y_46_re <= -1.85e-127) {
		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_re <= 4.2e+31) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (y_46_re / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.05e+76) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	} else if (y_46_re <= -1.85e-127) {
		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_re <= 4.2e+31) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (y_46_re / Math.hypot(y_46_im, y_46_re)) * (x_46_im / Math.hypot(y_46_im, y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.05e+76:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (((y_46_im * x_46_re) / y_46_re) - x_46_im)
	elif y_46_re <= -1.85e-127:
		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_re <= 4.2e+31:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = (y_46_re / math.hypot(y_46_im, y_46_re)) * (x_46_im / math.hypot(y_46_im, y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.05e+76)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) - x_46_im));
	elseif (y_46_re <= -1.85e-127)
		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_re <= 4.2e+31)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_im, y_46_re)) * Float64(x_46_im / hypot(y_46_im, y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.05e+76)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	elseif (y_46_re <= -1.85e-127)
		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_re <= 4.2e+31)
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = (y_46_re / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.05e+76], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.85e-127], 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$46$re, 4.2e+31], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.05000000000000003e76

   1. Initial program 42.5%

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

      1. *-un-lft-identity 42.5%

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

      2. add-sqr-sqrt 42.5%

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

      3. times-frac 42.5%

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

      4. hypot-define 42.5%

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

      5. fma-neg 42.5%

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

      6. distribute-rgt-neg-in 42.5%

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

      7. hypot-define 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in y.re around -inf 88.1%

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

   if -1.05000000000000003e76 < y.re < -1.8500000000000002e-127

   1. Initial program 79.5%

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

   if -1.8500000000000002e-127 < y.re < 4.19999999999999958e31

   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. Taylor expanded in y.re around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. mul-1-neg 81.7%

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

      3. unsub-neg 81.7%

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

      4. *-commutative 81.7%

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

      5. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. pow2 81.0%

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

      3. times-frac 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. associate-*r* 84.8%

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

      2. clear-num 84.8%

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

      3. un-div-inv 84.8%

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

      4. un-div-inv 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. div-inv 84.8%

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

       3. times-frac 89.3%

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

   11. Applied egg-rr 89.3%

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

   if 4.19999999999999958e31 < y.re

   1. Initial program 49.5%

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

   3. Taylor expanded in x.im around inf 46.3%

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

      1. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. add-sqr-sqrt 46.3%

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

      2. hypot-undefine 46.3%

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

      3. hypot-undefine 46.3%

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

      4. frac-times 84.0%

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

      5. hypot-undefine 47.9%

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

      6. +-commutative 47.9%

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

      7. hypot-define 84.0%

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

      8. hypot-undefine 47.9%

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

      9. +-commutative 47.9%

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

      10. hypot-define 84.0%

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

   7. Applied egg-rr 84.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.im \cdot x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{-127}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.65 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.3e+68)
   (- (/ x.im y.re) (* y.im (* (/ 1.0 y.re) (/ x.re y.re))))
   (if (<= y.re -1.65e-126)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.42e+44)
       (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im))
       (* (/ 1.0 (hypot y.re y.im)) (- x.im (* x.re (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.3e+68) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	} else if (y_46_re <= -1.65e-126) {
		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_re <= 1.42e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.3e+68) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	} else if (y_46_re <= -1.65e-126) {
		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_re <= 1.42e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.3e+68:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)))
	elif y_46_re <= -1.65e-126:
		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_re <= 1.42e+44:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.3e+68)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(1.0 / y_46_re) * Float64(x_46_re / y_46_re))));
	elseif (y_46_re <= -1.65e-126)
		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_re <= 1.42e+44)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.3e+68)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	elseif (y_46_re <= -1.65e-126)
		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_re <= 1.42e+44)
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.3e+68], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.65e-126], 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$46$re, 1.42e+44], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.2999999999999999e68

   1. Initial program 44.3%

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

   3. Taylor expanded in y.re around inf 82.8%

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

      1. +-commutative 82.8%

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

      2. mul-1-neg 82.8%

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

      3. unsub-neg 82.8%

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

      4. *-commutative 82.8%

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

      5. associate-/l* 83.0%

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

   5. Simplified 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. pow2 83.0%

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

      3. times-frac 84.6%

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

   7. Applied egg-rr 84.6%

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

   if -1.2999999999999999e68 < y.re < -1.65e-126

   1. Initial program 78.4%

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

   if -1.65e-126 < y.re < 1.41999999999999994e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

      4. un-div-inv 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. *-un-lft-identity 84.9%

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

       2. div-inv 84.9%

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

       3. times-frac 89.4%

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

   11. Applied egg-rr 89.4%

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

   if 1.41999999999999994e44 < y.re

   1. Initial program 48.5%

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

      1. *-un-lft-identity 48.5%

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

      2. add-sqr-sqrt 48.5%

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

      3. times-frac 48.5%

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

      4. hypot-define 48.5%

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

      5. fma-neg 48.5%

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

      6. distribute-rgt-neg-in 48.5%

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

      7. hypot-define 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in y.re around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

      3. associate-/l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.65 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.42 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := x.re \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+77}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x.im - t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (* x.re (/ y.im y.re))))
   (if (<= y.re -3.7e+77)
     (* t_0 (- t_1 x.im))
     (if (<= y.re -5.3e-127)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.85e+44)
         (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im))
         (* t_0 (- x.im t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -3.7e+77) {
		tmp = t_0 * (t_1 - x_46_im);
	} else if (y_46_re <= -5.3e-127) {
		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_re <= 1.85e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0 * (x_46_im - t_1);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = x_46_re * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -3.7e+77) {
		tmp = t_0 * (t_1 - x_46_im);
	} else if (y_46_re <= -5.3e-127) {
		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_re <= 1.85e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0 * (x_46_im - t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = x_46_re * (y_46_im / y_46_re)
	tmp = 0
	if y_46_re <= -3.7e+77:
		tmp = t_0 * (t_1 - x_46_im)
	elif y_46_re <= -5.3e-127:
		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_re <= 1.85e+44:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = t_0 * (x_46_im - t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_re * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.7e+77)
		tmp = Float64(t_0 * Float64(t_1 - x_46_im));
	elseif (y_46_re <= -5.3e-127)
		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_re <= 1.85e+44)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(t_0 * Float64(x_46_im - t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = x_46_re * (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.7e+77)
		tmp = t_0 * (t_1 - x_46_im);
	elseif (y_46_re <= -5.3e-127)
		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_re <= 1.85e+44)
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = t_0 * (x_46_im - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.7e+77], N[(t$95$0 * N[(t$95$1 - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.3e-127], 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$46$re, 1.85e+44], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$46$im - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.69999999999999995e77

   1. Initial program 42.5%

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

      1. *-un-lft-identity 42.5%

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

      2. add-sqr-sqrt 42.5%

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

      3. times-frac 42.5%

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

      4. hypot-define 42.5%

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

      5. fma-neg 42.5%

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

      6. distribute-rgt-neg-in 42.5%

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

      7. hypot-define 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in y.re around -inf 88.1%

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

      1. neg-mul-1 88.1%

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

      2. +-commutative 88.1%

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

      3. unsub-neg 88.1%

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

      4. associate-/l* 87.3%

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

   7. Simplified 87.3%

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

   if -3.69999999999999995e77 < y.re < -5.3000000000000003e-127

   1. Initial program 79.5%

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

   if -5.3000000000000003e-127 < y.re < 1.85e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

      4. un-div-inv 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. *-un-lft-identity 84.9%

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

       2. div-inv 84.9%

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

       3. times-frac 89.4%

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

   11. Applied egg-rr 89.4%

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

   if 1.85e44 < y.re

   1. Initial program 48.5%

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

      1. *-un-lft-identity 48.5%

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

      2. add-sqr-sqrt 48.5%

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

      3. times-frac 48.5%

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

      4. hypot-define 48.5%

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

      5. fma-neg 48.5%

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

      6. distribute-rgt-neg-in 48.5%

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

      7. hypot-define 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in y.re around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

      3. associate-/l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot \frac{y.im}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -5.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0 \cdot \left(\frac{y.im \cdot x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))))
   (if (<= y.re -3.8e+77)
     (* t_0 (- (/ (* y.im x.re) y.re) x.im))
     (if (<= y.re -1.6e-126)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 3.45e+44)
         (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im))
         (* t_0 (- x.im (* x.re (/ y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -3.8e+77) {
		tmp = t_0 * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	} else if (y_46_re <= -1.6e-126) {
		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_re <= 3.45e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0 * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -3.8e+77) {
		tmp = t_0 * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	} else if (y_46_re <= -1.6e-126) {
		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_re <= 3.45e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0 * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	tmp = 0
	if y_46_re <= -3.8e+77:
		tmp = t_0 * (((y_46_im * x_46_re) / y_46_re) - x_46_im)
	elif y_46_re <= -1.6e-126:
		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_re <= 3.45e+44:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = t_0 * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -3.8e+77)
		tmp = Float64(t_0 * Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) - x_46_im));
	elseif (y_46_re <= -1.6e-126)
		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_re <= 3.45e+44)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(t_0 * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	tmp = 0.0;
	if (y_46_re <= -3.8e+77)
		tmp = t_0 * (((y_46_im * x_46_re) / y_46_re) - x_46_im);
	elseif (y_46_re <= -1.6e-126)
		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_re <= 3.45e+44)
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = t_0 * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.8e+77], N[(t$95$0 * N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.6e-126], 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$46$re, 3.45e+44], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.8000000000000001e77

   1. Initial program 42.5%

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

      1. *-un-lft-identity 42.5%

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

      2. add-sqr-sqrt 42.5%

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

      3. times-frac 42.5%

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

      4. hypot-define 42.5%

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

      5. fma-neg 42.5%

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

      6. distribute-rgt-neg-in 42.5%

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

      7. hypot-define 65.0%

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

   4. Applied egg-rr 65.0%

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

   5. Taylor expanded in y.re around -inf 88.1%

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

   if -3.8000000000000001e77 < y.re < -1.6e-126

   1. Initial program 79.5%

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

   if -1.6e-126 < y.re < 3.4499999999999999e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

      4. un-div-inv 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. *-un-lft-identity 84.9%

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

       2. div-inv 84.9%

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

       3. times-frac 89.4%

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

   11. Applied egg-rr 89.4%

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

   if 3.4499999999999999e44 < y.re

   1. Initial program 48.5%

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

      1. *-un-lft-identity 48.5%

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

      2. add-sqr-sqrt 48.5%

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

      3. times-frac 48.5%

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

      4. hypot-define 48.5%

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

      5. fma-neg 48.5%

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

      6. distribute-rgt-neg-in 48.5%

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

      7. hypot-define 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in y.re around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

      3. associate-/l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.im \cdot x.re}{y.re} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.1 \cdot 10^{-6}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(y.im \cdot x.re\right) \cdot \frac{-1}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.25e+65)
   (/ x.im y.re)
   (if (<= y.re -4.1e-6)
     (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
     (if (<= y.re -5.8e-92)
       (/ x.im y.re)
       (if (<= y.re -9.5e-126)
         (/ (* (* y.im x.re) (/ -1.0 y.re)) y.re)
         (if (<= y.re 1.75e+44)
           (- (* x.im (/ (/ y.re y.im) y.im)) (/ x.re y.im))
           (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.25e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.1e-6) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.8e-92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.5e-126) {
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	} else if (y_46_re <= 1.75e+44) {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.25d+65)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-4.1d-6)) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= (-5.8d-92)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-9.5d-126)) then
        tmp = ((y_46im * x_46re) * ((-1.0d0) / y_46re)) / y_46re
    else if (y_46re <= 1.75d+44) then
        tmp = (x_46im * ((y_46re / y_46im) / y_46im)) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.25e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.1e-6) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.8e-92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.5e-126) {
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	} else if (y_46_re <= 1.75e+44) {
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.25e+65:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -4.1e-6:
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= -5.8e-92:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -9.5e-126:
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re
	elif y_46_re <= 1.75e+44:
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.25e+65)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -4.1e-6)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= -5.8e-92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -9.5e-126)
		tmp = Float64(Float64(Float64(y_46_im * x_46_re) * Float64(-1.0 / y_46_re)) / y_46_re);
	elseif (y_46_re <= 1.75e+44)
		tmp = Float64(Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.25e+65)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -4.1e-6)
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= -5.8e-92)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -9.5e-126)
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	elseif (y_46_re <= 1.75e+44)
		tmp = (x_46_im * ((y_46_re / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.25e+65], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.1e-6], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.8e-92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -9.5e-126], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.75e+44], N[(N[(x$46$im * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.24999999999999993e65 or -4.0999999999999997e-6 < y.re < -5.79999999999999969e-92 or 1.75e44 < y.re

   1. Initial program 49.5%

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

   3. Taylor expanded in y.re around inf 74.5%

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

   if -1.24999999999999993e65 < y.re < -4.0999999999999997e-6

   1. Initial program 69.2%

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

   3. Taylor expanded in y.re around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. mul-1-neg 51.8%

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

      3. unsub-neg 51.8%

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

      4. *-commutative 51.8%

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

      5. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. pow2 63.8%

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

      3. times-frac 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. associate-*l/ 63.8%

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

      2. *-lft-identity 63.8%

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

   9. Simplified 63.8%

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

   if -5.79999999999999969e-92 < y.re < -9.5000000000000003e-126

   1. Initial program 100.0%

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

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

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

      1. +-commutative 72.2%

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

      2. mul-1-neg 72.2%

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

      3. unsub-neg 72.2%

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

      4. *-commutative 72.2%

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

      5. associate-/l* 32.0%

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

   5. Simplified 32.0%

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

   6. Taylor expanded in x.im around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*r/ 18.0%

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

      4. distribute-rgt-neg-out 18.0%

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

      5. distribute-frac-neg2 18.0%

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

   8. Simplified 18.0%

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

      1. associate-*r/ 58.3%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-/r* 0.0%

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

   10. Applied egg-rr 1.2%

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

       1. add-sqr-sqrt 0.7%

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

       2. sqrt-unprod 30.6%

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

       3. sqr-neg 30.6%

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

       4. mul-1-neg 30.6%

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

       5. mul-1-neg 30.6%

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

       6. sqrt-unprod 43.2%

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

       7. add-sqr-sqrt 58.1%

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

       8. mul-1-neg 58.1%

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

       9. div-inv 58.3%

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

       10. distribute-rgt-neg-in 58.3%

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

       11. *-commutative 58.3%

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

   12. Applied egg-rr 58.3%

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

   if -9.5000000000000003e-126 < y.re < 1.75e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

      4. un-div-inv 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. associate-/r/ 88.4%

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

   11. Applied egg-rr 88.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.1 \cdot 10^{-6}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(y.im \cdot x.re\right) \cdot \frac{-1}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+44}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(y.im \cdot x.re\right) \cdot \frac{-1}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.65e+65)
   (/ x.im y.re)
   (if (<= y.re -1.2e-9)
     (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
     (if (<= y.re -5.2e-92)
       (/ x.im y.re)
       (if (<= y.re -9.5e-126)
         (/ (* (* y.im x.re) (/ -1.0 y.re)) y.re)
         (if (<= y.re 8.8e+44)
           (- (/ (* (/ y.re y.im) x.im) y.im) (/ x.re y.im))
           (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.65e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.2e-9) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.2e-92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.5e-126) {
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	} else if (y_46_re <= 8.8e+44) {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.65d+65)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-1.2d-9)) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= (-5.2d-92)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-9.5d-126)) then
        tmp = ((y_46im * x_46re) * ((-1.0d0) / y_46re)) / y_46re
    else if (y_46re <= 8.8d+44) then
        tmp = (((y_46re / y_46im) * x_46im) / y_46im) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.65e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -1.2e-9) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.2e-92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -9.5e-126) {
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	} else if (y_46_re <= 8.8e+44) {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.65e+65:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -1.2e-9:
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= -5.2e-92:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -9.5e-126:
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re
	elif y_46_re <= 8.8e+44:
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.65e+65)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -1.2e-9)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= -5.2e-92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -9.5e-126)
		tmp = Float64(Float64(Float64(y_46_im * x_46_re) * Float64(-1.0 / y_46_re)) / y_46_re);
	elseif (y_46_re <= 8.8e+44)
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.65e+65)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -1.2e-9)
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= -5.2e-92)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -9.5e-126)
		tmp = ((y_46_im * x_46_re) * (-1.0 / y_46_re)) / y_46_re;
	elseif (y_46_re <= 8.8e+44)
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.65e+65], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.2e-9], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -9.5e-126], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 8.8e+44], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.65000000000000012e65 or -1.2e-9 < y.re < -5.2e-92 or 8.79999999999999983e44 < y.re

   1. Initial program 49.5%

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

   3. Taylor expanded in y.re around inf 74.5%

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

   if -1.65000000000000012e65 < y.re < -1.2e-9

   1. Initial program 69.2%

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

   3. Taylor expanded in y.re around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. mul-1-neg 51.8%

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

      3. unsub-neg 51.8%

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

      4. *-commutative 51.8%

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

      5. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. pow2 63.8%

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

      3. times-frac 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. associate-*l/ 63.8%

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

      2. *-lft-identity 63.8%

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

   9. Simplified 63.8%

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

   if -5.2e-92 < y.re < -9.5000000000000003e-126

   1. Initial program 100.0%

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

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

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

      1. +-commutative 72.2%

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

      2. mul-1-neg 72.2%

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

      3. unsub-neg 72.2%

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

      4. *-commutative 72.2%

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

      5. associate-/l* 32.0%

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

   5. Simplified 32.0%

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

   6. Taylor expanded in x.im around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*r/ 18.0%

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

      4. distribute-rgt-neg-out 18.0%

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

      5. distribute-frac-neg2 18.0%

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

   8. Simplified 18.0%

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

      1. associate-*r/ 58.3%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-/r* 0.0%

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

   10. Applied egg-rr 1.2%

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

       1. add-sqr-sqrt 0.7%

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

       2. sqrt-unprod 30.6%

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

       3. sqr-neg 30.6%

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

       4. mul-1-neg 30.6%

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

       5. mul-1-neg 30.6%

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

       6. sqrt-unprod 43.2%

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

       7. add-sqr-sqrt 58.1%

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

       8. mul-1-neg 58.1%

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

       9. div-inv 58.3%

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

       10. distribute-rgt-neg-in 58.3%

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

       11. *-commutative 58.3%

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

   12. Applied egg-rr 58.3%

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

   if -9.5000000000000003e-126 < y.re < 8.79999999999999983e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. associate-*r/ 89.4%

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

      3. un-div-inv 89.4%

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

   9. Applied egg-rr 89.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left(y.im \cdot x.re\right) \cdot \frac{-1}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -3050000:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98} \lor \neg \left(y.re \leq 1.28 \cdot 10^{+46}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (* y.im (* (/ 1.0 y.re) (/ x.re y.re))))))
   (if (<= y.re -1.3e+65)
     t_0
     (if (<= y.re -3050000.0)
       (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
       (if (or (<= y.re -1.2e-98) (not (<= y.re 1.28e+46)))
         t_0
         (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -1.3e+65) {
		tmp = t_0;
	} else if (y_46_re <= -3050000.0) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if ((y_46_re <= -1.2e-98) || !(y_46_re <= 1.28e+46)) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - (y_46im * ((1.0d0 / y_46re) * (x_46re / y_46re)))
    if (y_46re <= (-1.3d+65)) then
        tmp = t_0
    else if (y_46re <= (-3050000.0d0)) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else if ((y_46re <= (-1.2d-98)) .or. (.not. (y_46re <= 1.28d+46))) then
        tmp = t_0
    else
        tmp = ((1.0d0 / y_46im) * ((y_46re / y_46im) / (1.0d0 / 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 t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -1.3e+65) {
		tmp = t_0;
	} else if (y_46_re <= -3050000.0) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if ((y_46_re <= -1.2e-98) || !(y_46_re <= 1.28e+46)) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)))
	tmp = 0
	if y_46_re <= -1.3e+65:
		tmp = t_0
	elif y_46_re <= -3050000.0:
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	elif (y_46_re <= -1.2e-98) or not (y_46_re <= 1.28e+46):
		tmp = t_0
	else:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(1.0 / y_46_re) * Float64(x_46_re / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.3e+65)
		tmp = t_0;
	elseif (y_46_re <= -3050000.0)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif ((y_46_re <= -1.2e-98) || !(y_46_re <= 1.28e+46))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.3e+65)
		tmp = t_0;
	elseif (y_46_re <= -3050000.0)
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	elseif ((y_46_re <= -1.2e-98) || ~((y_46_re <= 1.28e+46)))
		tmp = t_0;
	else
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e+65], t$95$0, If[LessEqual[y$46$re, -3050000.0], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -1.2e-98], N[Not[LessEqual[y$46$re, 1.28e+46]], $MachinePrecision]], t$95$0, N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.30000000000000001e65 or -3.05e6 < y.re < -1.20000000000000002e-98 or 1.2800000000000001e46 < y.re

   1. Initial program 51.0%

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

   3. Taylor expanded in y.re around inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. pow2 77.6%

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

      3. times-frac 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -1.30000000000000001e65 < y.re < -3.05e6

   1. Initial program 64.9%

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

   3. Taylor expanded in y.re around 0 58.5%

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

      1. +-commutative 58.5%

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

      2. mul-1-neg 58.5%

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

      3. unsub-neg 58.5%

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

      4. *-commutative 58.5%

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

      5. associate-/l* 72.2%

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

   5. Simplified 72.2%

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

      1. *-un-lft-identity 72.2%

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

      2. pow2 72.2%

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

      3. times-frac 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. associate-*l/ 72.2%

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

      2. *-lft-identity 72.2%

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

   9. Simplified 72.2%

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

   if -1.20000000000000002e-98 < y.re < 1.2800000000000001e46

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. mul-1-neg 80.0%

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

      3. unsub-neg 80.0%

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

      4. *-commutative 80.0%

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

      5. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. *-un-lft-identity 79.4%

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

      2. pow2 79.4%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r* 82.9%

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

      2. clear-num 83.0%

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

      3. un-div-inv 83.0%

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

      4. un-div-inv 83.0%

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

   9. Applied egg-rr 83.0%

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

       1. *-un-lft-identity 83.0%

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

       2. div-inv 82.9%

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

       3. times-frac 87.2%

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

   11. Applied egg-rr 87.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -3050000:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98} \lor \neg \left(y.re \leq 1.28 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98} \lor \neg \left(y.re \leq 2.9 \cdot 10^{+45}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (* y.im (* (/ 1.0 y.re) (/ x.re y.re))))))
   (if (<= y.re -7.5e+65)
     t_0
     (if (<= y.re -3.2e+16)
       (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
       (if (or (<= y.re -1.2e-98) (not (<= y.re 2.9e+45)))
         t_0
         (- (/ (* (/ y.re y.im) x.im) y.im) (/ x.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -7.5e+65) {
		tmp = t_0;
	} else if (y_46_re <= -3.2e+16) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if ((y_46_re <= -1.2e-98) || !(y_46_re <= 2.9e+45)) {
		tmp = t_0;
	} else {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - (y_46im * ((1.0d0 / y_46re) * (x_46re / y_46re)))
    if (y_46re <= (-7.5d+65)) then
        tmp = t_0
    else if (y_46re <= (-3.2d+16)) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else if ((y_46re <= (-1.2d-98)) .or. (.not. (y_46re <= 2.9d+45))) then
        tmp = t_0
    else
        tmp = (((y_46re / y_46im) * x_46im) / y_46im) - (x_46re / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -7.5e+65) {
		tmp = t_0;
	} else if (y_46_re <= -3.2e+16) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if ((y_46_re <= -1.2e-98) || !(y_46_re <= 2.9e+45)) {
		tmp = t_0;
	} else {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)))
	tmp = 0
	if y_46_re <= -7.5e+65:
		tmp = t_0
	elif y_46_re <= -3.2e+16:
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	elif (y_46_re <= -1.2e-98) or not (y_46_re <= 2.9e+45):
		tmp = t_0
	else:
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(1.0 / y_46_re) * Float64(x_46_re / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -7.5e+65)
		tmp = t_0;
	elseif (y_46_re <= -3.2e+16)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif ((y_46_re <= -1.2e-98) || !(y_46_re <= 2.9e+45))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -7.5e+65)
		tmp = t_0;
	elseif (y_46_re <= -3.2e+16)
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	elseif ((y_46_re <= -1.2e-98) || ~((y_46_re <= 2.9e+45)))
		tmp = t_0;
	else
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+65], t$95$0, If[LessEqual[y$46$re, -3.2e+16], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -1.2e-98], N[Not[LessEqual[y$46$re, 2.9e+45]], $MachinePrecision]], t$95$0, N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000006e65 or -3.2e16 < y.re < -1.20000000000000002e-98 or 2.8999999999999997e45 < y.re

   1. Initial program 51.0%

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

   3. Taylor expanded in y.re around inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. pow2 77.6%

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

      3. times-frac 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -7.50000000000000006e65 < y.re < -3.2e16

   1. Initial program 64.9%

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

   3. Taylor expanded in y.re around 0 58.5%

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

      1. +-commutative 58.5%

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

      2. mul-1-neg 58.5%

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

      3. unsub-neg 58.5%

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

      4. *-commutative 58.5%

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

      5. associate-/l* 72.2%

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

   5. Simplified 72.2%

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

      1. *-un-lft-identity 72.2%

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

      2. pow2 72.2%

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

      3. times-frac 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. associate-*l/ 72.2%

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

      2. *-lft-identity 72.2%

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

   9. Simplified 72.2%

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

   if -1.20000000000000002e-98 < y.re < 2.8999999999999997e45

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. mul-1-neg 80.0%

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

      3. unsub-neg 80.0%

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

      4. *-commutative 80.0%

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

      5. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. *-un-lft-identity 79.4%

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

      2. pow2 79.4%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r* 82.9%

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

      2. associate-*r/ 87.2%

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

      3. un-div-inv 87.2%

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

   9. Applied egg-rr 87.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98} \lor \neg \left(y.re \leq 2.9 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -0.000235:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.6e+65)
   (/ x.im y.re)
   (if (<= y.re -0.000235)
     (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
     (if (<= y.re -1.2e-98)
       (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 2.25e+45)
         (- (/ (* (/ y.re y.im) x.im) y.im) (/ x.re y.im))
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.6e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -0.000235) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -1.2e-98) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.25e+45) {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-7.6d+65)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-0.000235d0)) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else if (y_46re <= (-1.2d-98)) then
        tmp = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 2.25d+45) then
        tmp = (((y_46re / y_46im) * x_46im) / y_46im) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.6e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -0.000235) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else if (y_46_re <= -1.2e-98) {
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 2.25e+45) {
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.6e+65:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -0.000235:
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	elif y_46_re <= -1.2e-98:
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 2.25e+45:
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.6e+65)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -0.000235)
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= -1.2e-98)
		tmp = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 2.25e+45)
		tmp = Float64(Float64(Float64(Float64(y_46_re / y_46_im) * x_46_im) / y_46_im) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.6e+65)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -0.000235)
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	elseif (y_46_re <= -1.2e-98)
		tmp = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 2.25e+45)
		tmp = (((y_46_re / y_46_im) * x_46_im) / y_46_im) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.6e+65], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -0.000235], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.2e-98], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.25e+45], N[(N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.60000000000000022e65 or 2.2499999999999999e45 < y.re

   1. Initial program 46.1%

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

   3. Taylor expanded in y.re around inf 76.1%

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

   if -7.60000000000000022e65 < y.re < -2.34999999999999993e-4

   1. Initial program 69.2%

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

   3. Taylor expanded in y.re around 0 51.8%

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

      1. +-commutative 51.8%

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

      2. mul-1-neg 51.8%

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

      3. unsub-neg 51.8%

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

      4. *-commutative 51.8%

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

      5. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. pow2 63.8%

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

      3. times-frac 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. associate-*l/ 63.8%

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

      2. *-lft-identity 63.8%

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

   9. Simplified 63.8%

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

   if -2.34999999999999993e-4 < y.re < -1.20000000000000002e-98

   1. Initial program 80.9%

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

   3. Taylor expanded in x.im around inf 62.5%

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

      1. *-commutative 62.5%

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

   5. Simplified 62.5%

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

   if -1.20000000000000002e-98 < y.re < 2.2499999999999999e45

   1. Initial program 74.4%

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

   3. Taylor expanded in y.re around 0 80.0%

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

      1. +-commutative 80.0%

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

      2. mul-1-neg 80.0%

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

      3. unsub-neg 80.0%

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

      4. *-commutative 80.0%

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

      5. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. *-un-lft-identity 79.4%

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

      2. pow2 79.4%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r* 82.9%

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

      2. associate-*r/ 87.2%

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

      3. un-div-inv 87.2%

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

   9. Applied egg-rr 87.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -0.000235:\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{y.re}{y.im} \cdot x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.76 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im y.re) (* y.im (* (/ 1.0 y.re) (/ x.re y.re))))))
   (if (<= y.re -1.7e+68)
     t_0
     (if (<= y.re -6.6e-126)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 1.76e+44)
         (- (* (/ 1.0 y.im) (/ (/ y.re y.im) (/ 1.0 x.im))) (/ x.re y.im))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -1.7e+68) {
		tmp = t_0;
	} else if (y_46_re <= -6.6e-126) {
		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_re <= 1.76e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46re) - (y_46im * ((1.0d0 / y_46re) * (x_46re / y_46re)))
    if (y_46re <= (-1.7d+68)) then
        tmp = t_0
    else if (y_46re <= (-6.6d-126)) then
        tmp = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 1.76d+44) then
        tmp = ((1.0d0 / y_46im) * ((y_46re / y_46im) / (1.0d0 / x_46im))) - (x_46re / y_46im)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -1.7e+68) {
		tmp = t_0;
	} else if (y_46_re <= -6.6e-126) {
		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_re <= 1.76e+44) {
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)))
	tmp = 0
	if y_46_re <= -1.7e+68:
		tmp = t_0
	elif y_46_re <= -6.6e-126:
		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_re <= 1.76e+44:
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(1.0 / y_46_re) * Float64(x_46_re / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.7e+68)
		tmp = t_0;
	elseif (y_46_re <= -6.6e-126)
		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_re <= 1.76e+44)
		tmp = Float64(Float64(Float64(1.0 / y_46_im) * Float64(Float64(y_46_re / y_46_im) / Float64(1.0 / x_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_re) - (y_46_im * ((1.0 / y_46_re) * (x_46_re / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.7e+68)
		tmp = t_0;
	elseif (y_46_re <= -6.6e-126)
		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_re <= 1.76e+44)
		tmp = ((1.0 / y_46_im) * ((y_46_re / y_46_im) / (1.0 / x_46_im))) - (x_46_re / y_46_im);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.7e+68], t$95$0, If[LessEqual[y$46$re, -6.6e-126], 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$46$re, 1.76e+44], N[(N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(1.0 / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.70000000000000008e68 or 1.76e44 < y.re

   1. Initial program 46.1%

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

   3. Taylor expanded in y.re around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. mul-1-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. *-commutative 78.2%

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

      5. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. pow2 78.6%

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

      3. times-frac 81.8%

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

   7. Applied egg-rr 81.8%

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

   if -1.70000000000000008e68 < y.re < -6.6000000000000001e-126

   1. Initial program 78.4%

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

   if -6.6000000000000001e-126 < y.re < 1.76e44

   1. Initial program 73.2%

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

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

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

      1. +-commutative 81.9%

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

      2. mul-1-neg 81.9%

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

      3. unsub-neg 81.9%

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

      4. *-commutative 81.9%

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

      5. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

      1. *-un-lft-identity 81.2%

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

      2. pow2 81.2%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

      1. associate-*r* 84.9%

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

      2. clear-num 85.0%

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

      3. un-div-inv 85.0%

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

      4. un-div-inv 84.9%

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

   9. Applied egg-rr 84.9%

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

       1. *-un-lft-identity 84.9%

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

       2. div-inv 84.9%

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

       3. times-frac 89.4%

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

   11. Applied egg-rr 89.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.76 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{y.im} \cdot \frac{\frac{y.re}{y.im}}{\frac{1}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \left(\frac{1}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+47} \lor \neg \left(y.re \leq -1.26 \cdot 10^{-6} \lor \neg \left(y.re \leq -1.2 \cdot 10^{-98}\right) \land y.re \leq 1.36 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.7e+47)
         (not
          (or (<= y.re -1.26e-6)
              (and (not (<= y.re -1.2e-98)) (<= y.re 1.36e+44)))))
   (/ x.im y.re)
   (/ x.re (- y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.7e+47) || !((y_46_re <= -1.26e-6) || (!(y_46_re <= -1.2e-98) && (y_46_re <= 1.36e+44)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.7d+47)) .or. (.not. (y_46re <= (-1.26d-6)) .or. (.not. (y_46re <= (-1.2d-98))) .and. (y_46re <= 1.36d+44))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.7e+47) || !((y_46_re <= -1.26e-6) || (!(y_46_re <= -1.2e-98) && (y_46_re <= 1.36e+44)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.7e+47) or not ((y_46_re <= -1.26e-6) or (not (y_46_re <= -1.2e-98) and (y_46_re <= 1.36e+44))):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.7e+47) || !((y_46_re <= -1.26e-6) || (!(y_46_re <= -1.2e-98) && (y_46_re <= 1.36e+44))))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.7e+47) || ~(((y_46_re <= -1.26e-6) || (~((y_46_re <= -1.2e-98)) && (y_46_re <= 1.36e+44)))))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.7e+47], N[Not[Or[LessEqual[y$46$re, -1.26e-6], And[N[Not[LessEqual[y$46$re, -1.2e-98]], $MachinePrecision], LessEqual[y$46$re, 1.36e+44]]]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.6999999999999999e47 or -1.26000000000000001e-6 < y.re < -1.20000000000000002e-98 or 1.36000000000000005e44 < y.re

   1. Initial program 51.4%

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

   3. Taylor expanded in y.re around inf 72.8%

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

   if -1.6999999999999999e47 < y.re < -1.26000000000000001e-6 or -1.20000000000000002e-98 < y.re < 1.36000000000000005e44

   1. Initial program 73.8%

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

   3. Taylor expanded in y.re around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. neg-mul-1 67.3%

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

   5. Simplified 67.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+47} \lor \neg \left(y.re \leq -1.26 \cdot 10^{-6} \lor \neg \left(y.re \leq -1.2 \cdot 10^{-98}\right) \land y.re \leq 1.36 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-60} \lor \neg \left(y.im \leq 9 \cdot 10^{-64}\right):\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.8e-60) (not (<= y.im 9e-64)))
   (- (* y.re (/ (/ x.im y.im) y.im)) (/ x.re y.im))
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.8e-60) || !(y_46_im <= 9e-64)) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.8d-60)) .or. (.not. (y_46im <= 9d-64))) then
        tmp = (y_46re * ((x_46im / y_46im) / y_46im)) - (x_46re / y_46im)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.8e-60) || !(y_46_im <= 9e-64)) {
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.8e-60) or not (y_46_im <= 9e-64):
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.8e-60) || !(y_46_im <= 9e-64))
		tmp = Float64(Float64(y_46_re * Float64(Float64(x_46_im / y_46_im) / y_46_im)) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.8e-60) || ~((y_46_im <= 9e-64)))
		tmp = (y_46_re * ((x_46_im / y_46_im) / y_46_im)) - (x_46_re / y_46_im);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.8e-60], N[Not[LessEqual[y$46$im, 9e-64]], $MachinePrecision]], N[(N[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8e-60 or 9.00000000000000019e-64 < y.im

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around 0 63.9%

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

      1. +-commutative 63.9%

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

      2. mul-1-neg 63.9%

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

      3. unsub-neg 63.9%

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

      4. *-commutative 63.9%

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

      5. associate-/l* 68.2%

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

   5. Simplified 68.2%

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

      1. *-un-lft-identity 68.2%

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

      2. pow2 68.2%

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

      3. times-frac 71.2%

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

   7. Applied egg-rr 71.2%

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

      1. associate-*l/ 71.2%

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

      2. *-lft-identity 71.2%

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

   9. Simplified 71.2%

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

   if -1.8e-60 < y.im < 9.00000000000000019e-64

   1. Initial program 70.3%

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

   3. Taylor expanded in y.re around inf 75.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-60} \lor \neg \left(y.im \leq 9 \cdot 10^{-64}\right):\\ \;\;\;\;y.re \cdot \frac{\frac{x.im}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.8% 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 46.8%

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

4. Final simplification 46.8%

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

Reproduce

herbie shell --seed 2024053 
(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))))