_divideComplex, imaginary part

Percentage Accurate: 61.1% → 97.3%

Time: 20.2s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - (y_46_im / (math.hypot(y_46_re, y_46_im) / x_46_re))) / math.hypot(y_46_re, y_46_im)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. *-un-lft-identity 63.5%

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

   2. add-sqr-sqrt 63.5%

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

   3. times-frac 63.4%

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

   4. hypot-def 63.5%

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

   5. hypot-def 76.4%

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

4. Applied egg-rr 76.4%

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

   1. associate-*l/ 76.6%

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

   2. *-un-lft-identity 76.6%

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

   3. *-commutative 76.6%

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

   4. *-commutative 76.6%

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

6. Applied egg-rr 76.6%

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

   1. div-sub 76.6%

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

   2. sub-neg 76.6%

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

8. Applied egg-rr 76.6%

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

   1. sub-neg 76.6%

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

   2. associate-/l* 84.9%

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

   3. associate-/l* 97.2%

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

10. Simplified 97.2%

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

11. Final simplification 97.2%

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

Alternative 2: 88.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+254}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+254)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (- x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+254) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+254)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+254], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.9999999999999994e253

   1. Initial program 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 97.0%

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

   4. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. *-commutative 97.3%

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

      4. *-commutative 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 9.9999999999999994e253 < (/.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 11.4%

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

      1. div-sub 9.4%

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

      2. sub-neg 9.4%

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

      3. *-commutative 9.4%

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

      4. add-sqr-sqrt 9.4%

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

      5. times-frac 16.3%

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

      6. fma-def 16.3%

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

      7. hypot-def 16.3%

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

      8. hypot-def 45.6%

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

      9. associate-/l* 58.1%

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

      10. add-sqr-sqrt 58.1%

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

      11. pow2 58.1%

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

      12. hypot-def 58.1%

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

   4. Applied egg-rr 58.1%

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

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

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+254}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{x.re}{\frac{y.re}{y.im}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))))
   (if (<= y.re -1.3e+95)
     (/ (- (/ x.re (/ y.re y.im)) x.im) (hypot y.re y.im))
     (if (<= y.re -4.4e-100)
       t_0
       (if (<= y.re 1.8e-145)
         t_1
         (if (<= y.re 5e-66)
           t_0
           (if (<= y.re 2.35e-10)
             t_1
             (* (/ 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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -1.3e+95) {
		tmp = ((x_46_re / (y_46_re / y_46_im)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.4e-100) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-145) {
		tmp = t_1;
	} else if (y_46_re <= 5e-66) {
		tmp = t_0;
	} else if (y_46_re <= 2.35e-10) {
		tmp = t_1;
	} 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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -1.3e+95) {
		tmp = ((x_46_re / (y_46_re / y_46_im)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.4e-100) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-145) {
		tmp = t_1;
	} else if (y_46_re <= 5e-66) {
		tmp = t_0;
	} else if (y_46_re <= 2.35e-10) {
		tmp = t_1;
	} 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):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_re <= -1.3e+95:
		tmp = ((x_46_re / (y_46_re / y_46_im)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -4.4e-100:
		tmp = t_0
	elif y_46_re <= 1.8e-145:
		tmp = t_1
	elif y_46_re <= 5e-66:
		tmp = t_0
	elif y_46_re <= 2.35e-10:
		tmp = t_1
	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)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.3e+95)
		tmp = Float64(Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -4.4e-100)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-145)
		tmp = t_1;
	elseif (y_46_re <= 5e-66)
		tmp = t_0;
	elseif (y_46_re <= 2.35e-10)
		tmp = t_1;
	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)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.3e+95)
		tmp = ((x_46_re / (y_46_re / y_46_im)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -4.4e-100)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-145)
		tmp = t_1;
	elseif (y_46_re <= 5e-66)
		tmp = t_0;
	elseif (y_46_re <= 2.35e-10)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e+95], N[(N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.4e-100], t$95$0, If[LessEqual[y$46$re, 1.8e-145], t$95$1, If[LessEqual[y$46$re, 5e-66], t$95$0, If[LessEqual[y$46$re, 2.35e-10], t$95$1, 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 < -1.29999999999999995e95

   1. Initial program 49.8%

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

      1. *-un-lft-identity 49.8%

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

      2. add-sqr-sqrt 49.8%

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

      3. times-frac 49.8%

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

      4. hypot-def 49.8%

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

      5. hypot-def 65.8%

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

   4. Applied egg-rr 65.8%

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

      1. associate-*l/ 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. *-commutative 65.7%

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

      4. *-commutative 65.7%

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

   6. Applied egg-rr 65.7%

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

   7. Taylor expanded in y.re around -inf 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 95.0%

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

   9. Simplified 95.0%

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

   if -1.29999999999999995e95 < y.re < -4.39999999999999978e-100 or 1.8e-145 < y.re < 4.99999999999999962e-66

   1. Initial program 93.2%

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

   if -4.39999999999999978e-100 < y.re < 1.8e-145 or 4.99999999999999962e-66 < y.re < 2.3500000000000002e-10

   1. Initial program 59.4%

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

   3. Taylor expanded in y.re around 0 77.5%

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

      1. +-commutative 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-/l* 77.7%

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

   5. Simplified 77.7%

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

   if 2.3500000000000002e-10 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in x.im around inf 43.8%

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

      1. *-commutative 43.8%

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

   5. Simplified 43.8%

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

      1. add-sqr-sqrt 43.8%

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

      2. hypot-udef 43.8%

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

      3. hypot-udef 43.8%

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

      4. times-frac 83.7%

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

      5. hypot-udef 48.1%

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

      6. +-commutative 48.1%

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

      7. hypot-def 83.7%

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

      8. hypot-udef 48.1%

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

      9. +-commutative 48.1%

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

      10. hypot-def 83.7%

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

   7. Applied egg-rr 83.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{x.re}{\frac{y.re}{y.im}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-66}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{-10}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.1× speedup

Math

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

FPCore

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_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):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_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)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, 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

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Inf)
		tmp = (t_0 / hypot(y_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));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$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 77.8%

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

      1. *-un-lft-identity 77.8%

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

      2. add-sqr-sqrt 77.8%

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

      3. times-frac 77.7%

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

      4. hypot-def 77.7%

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

      5. hypot-def 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.2%

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

      2. *-un-lft-identity 93.2%

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

      3. *-commutative 93.2%

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

      4. *-commutative 93.2%

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

   6. Applied egg-rr 93.2%

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

   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-udef 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-udef 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. times-frac 57.1%

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

      5. hypot-udef 3.8%

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

      6. +-commutative 3.8%

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

      7. hypot-def 57.1%

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

      8. hypot-udef 3.8%

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

      9. +-commutative 3.8%

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

      10. hypot-def 57.1%

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

   7. Applied egg-rr 57.1%

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

4. Final simplification 86.5%

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

Alternative 5: 79.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.7e+90)
     (- (/ x.im y.re) (/ y.im (/ 1.0 (/ (/ x.re y.re) y.re))))
     (if (<= y.re -6e-100)
       t_0
       (if (<= y.re 1.1e-146)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
         (if (<= y.re 3.4e+80)
           t_0
           (/ (- x.im (/ x.re (/ y.re y.im))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.7e+90) {
		tmp = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	} else if (y_46_re <= -6e-100) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-146) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 3.4e+80) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.7e+90) {
		tmp = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	} else if (y_46_re <= -6e-100) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-146) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 3.4e+80) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.7e+90:
		tmp = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)))
	elif y_46_re <= -6e-100:
		tmp = t_0
	elif y_46_re <= 1.1e-146:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 3.4e+80:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / math.hypot(y_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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.7e+90)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(1.0 / Float64(Float64(x_46_re / y_46_re) / y_46_re))));
	elseif (y_46_re <= -6e-100)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-146)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 3.4e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.7e+90)
		tmp = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	elseif (y_46_re <= -6e-100)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-146)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 3.4e+80)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / hypot(y_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[(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, -2.7e+90], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(1.0 / N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6e-100], t$95$0, If[LessEqual[y$46$re, 1.1e-146], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.4e+80], t$95$0, N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.7e90

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around inf 84.3%

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

      1. +-commutative 84.3%

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

      2. mul-1-neg 84.3%

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

      3. unsub-neg 84.3%

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

      4. *-commutative 84.3%

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

      5. associate-/l* 87.4%

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

   5. Simplified 87.4%

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

      1. add-sqr-sqrt 51.9%

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

      2. sqrt-div 51.9%

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

      3. unpow2 51.9%

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

      4. sqrt-prod 0.0%

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

      5. add-sqr-sqrt 51.9%

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

      6. sqrt-div 51.9%

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

      7. unpow2 51.9%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 54.1%

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

   7. Applied egg-rr 54.1%

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

      1. unpow2 54.1%

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

   9. Simplified 54.1%

      \[\leadsto \frac{x.im}{y.re} - \frac{y.im}{\color{blue}{{\left(\frac{y.re}{\sqrt{x.re}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 54.1%

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

       2. clear-num 54.1%

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

       3. clear-num 54.1%

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

       4. frac-times 54.1%

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

       5. metadata-eval 54.1%

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

   11. Applied egg-rr 54.1%

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

       1. associate-*r/ 54.2%

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

       2. associate-*l/ 54.2%

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

       3. rem-square-sqrt 92.4%

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

   13. Simplified 92.4%

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

   if -2.7e90 < y.re < -6.0000000000000001e-100 or 1.1e-146 < y.re < 3.39999999999999992e80

   1. Initial program 86.4%

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

   if -6.0000000000000001e-100 < y.re < 1.1e-146

   1. Initial program 59.3%

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

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

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

   if 3.39999999999999992e80 < y.re

   1. Initial program 31.2%

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

      1. *-un-lft-identity 31.2%

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

      2. add-sqr-sqrt 31.2%

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

      3. times-frac 31.2%

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

      4. hypot-def 31.2%

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

      5. hypot-def 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. associate-*l/ 50.1%

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

      2. *-un-lft-identity 50.1%

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

      3. *-commutative 50.1%

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

      4. *-commutative 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in y.re around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. associate-/l* 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{t_1 - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ x.re (/ y.re y.im))))
   (if (<= y.re -5.8e+88)
     (/ (- t_1 x.im) (hypot y.re y.im))
     (if (<= y.re -4.8e-100)
       t_0
       (if (<= y.re 3e-142)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
         (if (<= y.re 1.65e+81) t_0 (/ (- x.im t_1) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -5.8e+88) {
		tmp = (t_1 - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.8e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3e-142) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.65e+81) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_re / y_46_im);
	double tmp;
	if (y_46_re <= -5.8e+88) {
		tmp = (t_1 - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.8e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3e-142) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 1.65e+81) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - t_1) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re / (y_46_re / y_46_im)
	tmp = 0
	if y_46_re <= -5.8e+88:
		tmp = (t_1 - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -4.8e-100:
		tmp = t_0
	elif y_46_re <= 3e-142:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 1.65e+81:
		tmp = t_0
	else:
		tmp = (x_46_im - t_1) / math.hypot(y_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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re / Float64(y_46_re / y_46_im))
	tmp = 0.0
	if (y_46_re <= -5.8e+88)
		tmp = Float64(Float64(t_1 - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -4.8e-100)
		tmp = t_0;
	elseif (y_46_re <= 3e-142)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 1.65e+81)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - t_1) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_re / (y_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -5.8e+88)
		tmp = (t_1 - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -4.8e-100)
		tmp = t_0;
	elseif (y_46_re <= 3e-142)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 1.65e+81)
		tmp = t_0;
	else
		tmp = (x_46_im - t_1) / hypot(y_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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+88], N[(N[(t$95$1 - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.8e-100], t$95$0, If[LessEqual[y$46$re, 3e-142], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+81], t$95$0, N[(N[(x$46$im - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.7999999999999999e88

   1. Initial program 49.8%

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

      1. *-un-lft-identity 49.8%

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

      2. add-sqr-sqrt 49.8%

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

      3. times-frac 49.8%

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

      4. hypot-def 49.8%

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

      5. hypot-def 65.8%

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

   4. Applied egg-rr 65.8%

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

      1. associate-*l/ 65.7%

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

      2. *-un-lft-identity 65.7%

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

      3. *-commutative 65.7%

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

      4. *-commutative 65.7%

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

   6. Applied egg-rr 65.7%

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

   7. Taylor expanded in y.re around -inf 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 95.0%

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

   9. Simplified 95.0%

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

   if -5.7999999999999999e88 < y.re < -4.8000000000000005e-100 or 3.0000000000000001e-142 < y.re < 1.65e81

   1. Initial program 86.4%

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

   if -4.8000000000000005e-100 < y.re < 3.0000000000000001e-142

   1. Initial program 59.3%

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

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

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

   if 1.65e81 < y.re

   1. Initial program 31.2%

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

      1. *-un-lft-identity 31.2%

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

      2. add-sqr-sqrt 31.2%

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

      3. times-frac 31.2%

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

      4. hypot-def 31.2%

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

      5. hypot-def 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. associate-*l/ 50.1%

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

      2. *-un-lft-identity 50.1%

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

      3. *-commutative 50.1%

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

      4. *-commutative 50.1%

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in y.re around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. associate-/l* 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{x.re}{\frac{y.re}{y.im}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ y.im (/ 1.0 (/ (/ x.re y.re) y.re))))))
   (if (<= y.re -1.05e+86)
     t_1
     (if (<= y.re -6.2e-100)
       t_0
       (if (<= y.re 3.2e-144)
         (- (/ x.im (/ (pow y.im 2.0) y.re)) (/ x.re y.im))
         (if (<= y.re 4.2e+80) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -1.05e+86) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3.2e-144) {
		tmp = (x_46_im / (pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 4.2e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (y_46im / (1.0d0 / ((x_46re / y_46re) / y_46re)))
    if (y_46re <= (-1.05d+86)) then
        tmp = t_1
    else if (y_46re <= (-6.2d-100)) then
        tmp = t_0
    else if (y_46re <= 3.2d-144) then
        tmp = (x_46im / ((y_46im ** 2.0d0) / y_46re)) - (x_46re / y_46im)
    else if (y_46re <= 4.2d+80) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -1.05e+86) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3.2e-144) {
		tmp = (x_46_im / (Math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im);
	} else if (y_46_re <= 4.2e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)))
	tmp = 0
	if y_46_re <= -1.05e+86:
		tmp = t_1
	elif y_46_re <= -6.2e-100:
		tmp = t_0
	elif y_46_re <= 3.2e-144:
		tmp = (x_46_im / (math.pow(y_46_im, 2.0) / y_46_re)) - (x_46_re / y_46_im)
	elif y_46_re <= 4.2e+80:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(1.0 / Float64(Float64(x_46_re / y_46_re) / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.05e+86)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-100)
		tmp = t_0;
	elseif (y_46_re <= 3.2e-144)
		tmp = Float64(Float64(x_46_im / Float64((y_46_im ^ 2.0) / y_46_re)) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= 4.2e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.05e+86)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-100)
		tmp = t_0;
	elseif (y_46_re <= 3.2e-144)
		tmp = (x_46_im / ((y_46_im ^ 2.0) / y_46_re)) - (x_46_re / y_46_im);
	elseif (y_46_re <= 4.2e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(1.0 / N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e+86], t$95$1, If[LessEqual[y$46$re, -6.2e-100], t$95$0, If[LessEqual[y$46$re, 3.2e-144], N[(N[(x$46$im / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+80], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.0499999999999999e86 or 4.20000000000000003e80 < y.re

   1. Initial program 40.0%

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

   3. Taylor expanded in y.re around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-/l* 83.9%

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

   5. Simplified 83.9%

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

      1. add-sqr-sqrt 49.9%

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

      2. sqrt-div 49.9%

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

      3. unpow2 49.9%

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

      4. sqrt-prod 25.3%

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

      5. add-sqr-sqrt 49.9%

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

      6. sqrt-div 49.9%

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

      7. unpow2 49.9%

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

      8. sqrt-prod 26.5%

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

      9. add-sqr-sqrt 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. unpow2 52.2%

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

   9. Simplified 52.2%

      \[\leadsto \frac{x.im}{y.re} - \frac{y.im}{\color{blue}{{\left(\frac{y.re}{\sqrt{x.re}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 52.2%

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

       2. clear-num 52.2%

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

       3. clear-num 52.2%

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

       4. frac-times 52.2%

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

       5. metadata-eval 52.2%

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

   11. Applied egg-rr 52.2%

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

       1. associate-*r/ 52.2%

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

       2. associate-*l/ 52.2%

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

       3. rem-square-sqrt 87.5%

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

   13. Simplified 87.5%

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

   if -1.0499999999999999e86 < y.re < -6.1999999999999997e-100 or 3.19999999999999973e-144 < y.re < 4.20000000000000003e80

   1. Initial program 86.4%

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

   if -6.1999999999999997e-100 < y.re < 3.19999999999999973e-144

   1. Initial program 59.3%

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

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

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.im}{\frac{{y.im}^{2}}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ y.im (/ 1.0 (/ (/ x.re y.re) y.re))))))
   (if (<= y.re -2.5e+94)
     t_1
     (if (<= y.re -3.2e-156)
       t_0
       (if (<= y.re 1.1e-146)
         (/ (- x.re) y.im)
         (if (<= y.re 1.35e+81) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -2.5e+94) {
		tmp = t_1;
	} else if (y_46_re <= -3.2e-156) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-146) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.35e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (y_46im / (1.0d0 / ((x_46re / y_46re) / y_46re)))
    if (y_46re <= (-2.5d+94)) then
        tmp = t_1
    else if (y_46re <= (-3.2d-156)) then
        tmp = t_0
    else if (y_46re <= 1.1d-146) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 1.35d+81) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -2.5e+94) {
		tmp = t_1;
	} else if (y_46_re <= -3.2e-156) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-146) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.35e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)))
	tmp = 0
	if y_46_re <= -2.5e+94:
		tmp = t_1
	elif y_46_re <= -3.2e-156:
		tmp = t_0
	elif y_46_re <= 1.1e-146:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 1.35e+81:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(1.0 / Float64(Float64(x_46_re / y_46_re) / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.5e+94)
		tmp = t_1;
	elseif (y_46_re <= -3.2e-156)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-146)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.35e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.5e+94)
		tmp = t_1;
	elseif (y_46_re <= -3.2e-156)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-146)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 1.35e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(1.0 / N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e+94], t$95$1, If[LessEqual[y$46$re, -3.2e-156], t$95$0, If[LessEqual[y$46$re, 1.1e-146], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+81], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.50000000000000005e94 or 1.35e81 < y.re

   1. Initial program 40.0%

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

   3. Taylor expanded in y.re around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. mul-1-neg 80.9%

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

      3. unsub-neg 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-/l* 83.9%

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

   5. Simplified 83.9%

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

      1. add-sqr-sqrt 49.9%

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

      2. sqrt-div 49.9%

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

      3. unpow2 49.9%

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

      4. sqrt-prod 25.3%

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

      5. add-sqr-sqrt 49.9%

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

      6. sqrt-div 49.9%

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

      7. unpow2 49.9%

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

      8. sqrt-prod 26.5%

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

      9. add-sqr-sqrt 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. unpow2 52.2%

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

   9. Simplified 52.2%

      \[\leadsto \frac{x.im}{y.re} - \frac{y.im}{\color{blue}{{\left(\frac{y.re}{\sqrt{x.re}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 52.2%

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

       2. clear-num 52.2%

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

       3. clear-num 52.2%

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

       4. frac-times 52.2%

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

       5. metadata-eval 52.2%

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

   11. Applied egg-rr 52.2%

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

       1. associate-*r/ 52.2%

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

       2. associate-*l/ 52.2%

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

       3. rem-square-sqrt 87.5%

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

   13. Simplified 87.5%

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

   if -2.50000000000000005e94 < y.re < -3.19999999999999982e-156 or 1.1e-146 < y.re < 1.35e81

   1. Initial program 85.0%

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

   if -3.19999999999999982e-156 < y.re < 1.1e-146

   1. Initial program 57.2%

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

   3. Taylor expanded in y.re around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   5. Simplified 72.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+81}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -9.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (/ y.im (/ 1.0 (/ (/ x.re y.re) y.re))))))
   (if (<= y.re -1.6e+44)
     t_1
     (if (<= y.re -9.8e-138)
       t_0
       (if (<= y.re 2.15e-10)
         (/ (- x.re) y.im)
         (if (<= y.re 1.05e+80) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -1.6e+44) {
		tmp = t_1;
	} else if (y_46_re <= -9.8e-138) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e-10) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.05e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (y_46im / (1.0d0 / ((x_46re / y_46re) / y_46re)))
    if (y_46re <= (-1.6d+44)) then
        tmp = t_1
    else if (y_46re <= (-9.8d-138)) then
        tmp = t_0
    else if (y_46re <= 2.15d-10) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 1.05d+80) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	double tmp;
	if (y_46_re <= -1.6e+44) {
		tmp = t_1;
	} else if (y_46_re <= -9.8e-138) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e-10) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.05e+80) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)))
	tmp = 0
	if y_46_re <= -1.6e+44:
		tmp = t_1
	elif y_46_re <= -9.8e-138:
		tmp = t_0
	elif y_46_re <= 2.15e-10:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 1.05e+80:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(1.0 / Float64(Float64(x_46_re / y_46_re) / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.6e+44)
		tmp = t_1;
	elseif (y_46_re <= -9.8e-138)
		tmp = t_0;
	elseif (y_46_re <= 2.15e-10)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.05e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (y_46_im / (1.0 / ((x_46_re / y_46_re) / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.6e+44)
		tmp = t_1;
	elseif (y_46_re <= -9.8e-138)
		tmp = t_0;
	elseif (y_46_re <= 2.15e-10)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 1.05e+80)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(1.0 / N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e+44], t$95$1, If[LessEqual[y$46$re, -9.8e-138], t$95$0, If[LessEqual[y$46$re, 2.15e-10], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+80], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.60000000000000002e44 or 1.05000000000000001e80 < y.re

   1. Initial program 46.9%

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

   3. Taylor expanded in y.re around inf 79.2%

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

      1. +-commutative 79.2%

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

      2. mul-1-neg 79.2%

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

      3. unsub-neg 79.2%

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

      4. *-commutative 79.2%

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

      5. associate-/l* 81.9%

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

   5. Simplified 81.9%

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

      1. add-sqr-sqrt 49.9%

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

      2. sqrt-div 49.9%

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

      3. unpow2 49.9%

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

      4. sqrt-prod 21.9%

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

      5. add-sqr-sqrt 47.8%

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

      6. sqrt-div 47.8%

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

      7. unpow2 47.8%

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

      8. sqrt-prod 23.0%

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

      9. add-sqr-sqrt 51.9%

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

   7. Applied egg-rr 51.9%

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

      1. unpow2 51.9%

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

   9. Simplified 51.9%

      \[\leadsto \frac{x.im}{y.re} - \frac{y.im}{\color{blue}{{\left(\frac{y.re}{\sqrt{x.re}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 51.9%

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

       2. clear-num 51.9%

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

       3. clear-num 51.9%

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

       4. frac-times 51.9%

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

       5. metadata-eval 51.9%

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

   11. Applied egg-rr 51.9%

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

       1. associate-*r/ 51.9%

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

       2. associate-*l/ 51.9%

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

       3. rem-square-sqrt 84.9%

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

   13. Simplified 84.9%

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

   if -1.60000000000000002e44 < y.re < -9.80000000000000033e-138 or 2.15000000000000007e-10 < y.re < 1.05000000000000001e80

   1. Initial program 86.6%

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

   3. Taylor expanded in x.im around inf 72.1%

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

      1. *-commutative 72.1%

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

   5. Simplified 72.1%

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

   if -9.80000000000000033e-138 < y.re < 2.15000000000000007e-10

   1. Initial program 64.5%

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

   3. Taylor expanded in y.re around 0 68.5%

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \mathbf{elif}\;y.re \leq -9.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{\frac{1}{\frac{\frac{x.re}{y.re}}{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.8e+97)
     (/ x.im y.re)
     (if (<= y.re -8.6e-138)
       t_0
       (if (<= y.re 2.15e-10)
         (/ (- x.re) y.im)
         (if (<= y.re 1.12e+80) t_0 (/ x.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 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.8e+97) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -8.6e-138) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e-10) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.12e+80) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.8d+97)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-8.6d-138)) then
        tmp = t_0
    else if (y_46re <= 2.15d-10) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 1.12d+80) then
        tmp = t_0
    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 t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.8e+97) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -8.6e-138) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e-10) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.12e+80) {
		tmp = t_0;
	} 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):
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.8e+97:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -8.6e-138:
		tmp = t_0
	elif y_46_re <= 2.15e-10:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 1.12e+80:
		tmp = t_0
	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)
	t_0 = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.8e+97)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -8.6e-138)
		tmp = t_0;
	elseif (y_46_re <= 2.15e-10)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.12e+80)
		tmp = t_0;
	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)
	t_0 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.8e+97)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -8.6e-138)
		tmp = t_0;
	elseif (y_46_re <= 2.15e-10)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 1.12e+80)
		tmp = t_0;
	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_] := Block[{t$95$0 = 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, -2.8e+97], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -8.6e-138], t$95$0, If[LessEqual[y$46$re, 2.15e-10], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.12e+80], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.7999999999999999e97 or 1.12e80 < y.re

   1. Initial program 40.0%

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

   3. Taylor expanded in y.re around inf 82.3%

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

   if -2.7999999999999999e97 < y.re < -8.6000000000000001e-138 or 2.15000000000000007e-10 < y.re < 1.12e80

   1. Initial program 87.4%

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

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

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

      1. *-commutative 71.4%

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

   5. Simplified 71.4%

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

   if -8.6000000000000001e-138 < y.re < 2.15000000000000007e-10

   1. Initial program 64.5%

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

   3. Taylor expanded in y.re around 0 68.5%

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1 (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))))
   (if (<= y.im -2.2e+46)
     t_0
     (if (<= y.im 2.7e-51)
       t_1
       (if (<= y.im 6e-8)
         (/ (* y.im (- x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 6.5e+58) t_1 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_re / y_46_im;
	double t_1 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_im <= -2.2e+46) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e-51) {
		tmp = t_1;
	} else if (y_46_im <= 6e-8) {
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 6.5e+58) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re / y_46im)))
    if (y_46im <= (-2.2d+46)) then
        tmp = t_0
    else if (y_46im <= 2.7d-51) then
        tmp = t_1
    else if (y_46im <= 6d-8) then
        tmp = (y_46im * -x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 6.5d+58) then
        tmp = t_1
    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_re / y_46_im;
	double t_1 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_im <= -2.2e+46) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e-51) {
		tmp = t_1;
	} else if (y_46_im <= 6e-8) {
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 6.5e+58) {
		tmp = t_1;
	} 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_re / y_46_im
	t_1 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	tmp = 0
	if y_46_im <= -2.2e+46:
		tmp = t_0
	elif y_46_im <= 2.7e-51:
		tmp = t_1
	elif y_46_im <= 6e-8:
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 6.5e+58:
		tmp = t_1
	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_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))))
	tmp = 0.0
	if (y_46_im <= -2.2e+46)
		tmp = t_0;
	elseif (y_46_im <= 2.7e-51)
		tmp = t_1;
	elseif (y_46_im <= 6e-8)
		tmp = Float64(Float64(y_46_im * Float64(-x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 6.5e+58)
		tmp = t_1;
	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_re / y_46_im;
	t_1 = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	tmp = 0.0;
	if (y_46_im <= -2.2e+46)
		tmp = t_0;
	elseif (y_46_im <= 2.7e-51)
		tmp = t_1;
	elseif (y_46_im <= 6e-8)
		tmp = (y_46_im * -x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 6.5e+58)
		tmp = t_1;
	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[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+46], t$95$0, If[LessEqual[y$46$im, 2.7e-51], t$95$1, If[LessEqual[y$46$im, 6e-8], N[(N[(y$46$im * (-x$46$re)), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.5e+58], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.2e46 or 6.49999999999999998e58 < y.im

   1. Initial program 50.0%

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

   3. Taylor expanded in y.re around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. neg-mul-1 77.0%

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

   5. Simplified 77.0%

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

   if -2.2e46 < y.im < 2.6999999999999997e-51 or 5.99999999999999946e-8 < y.im < 6.49999999999999998e58

   1. Initial program 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. add-sqr-sqrt 71.0%

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

      3. times-frac 70.9%

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

      4. hypot-def 71.0%

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

      5. hypot-def 85.0%

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

   4. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.2%

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

      2. *-un-lft-identity 85.2%

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

      3. *-commutative 85.2%

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

      4. *-commutative 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in y.re around inf 66.5%

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

      1. +-commutative 66.5%

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

      2. mul-1-neg 66.5%

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

      3. unsub-neg 66.5%

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

      4. associate-/l* 66.2%

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

   9. Simplified 66.2%

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

       1. pow2 66.2%

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

       2. *-un-lft-identity 66.2%

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

       3. times-frac 72.5%

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

   11. Applied egg-rr 72.5%

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

   if 2.6999999999999997e-51 < y.im < 5.99999999999999946e-8

   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 x.im around 0 86.6%

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

      1. associate-*r* 86.6%

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

      2. neg-mul-1 86.6%

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

      3. *-commutative 86.6%

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

   5. Simplified 86.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.im \cdot \left(-x.re\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -29500 \lor \neg \left(y.re \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -29500.0) (not (<= y.re 1.35e+51)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -29500.0) || !(y_46_re <= 1.35e+51)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-29500.0d0)) .or. (.not. (y_46re <= 1.35d+51))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -29500.0) || !(y_46_re <= 1.35e+51)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -29500.0) or not (y_46_re <= 1.35e+51):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -29500.0) || !(y_46_re <= 1.35e+51))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -29500.0) || ~((y_46_re <= 1.35e+51)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -29500.0], N[Not[LessEqual[y$46$re, 1.35e+51]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -29500 or 1.34999999999999996e51 < y.re

   1. Initial program 52.6%

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

   3. Taylor expanded in y.re around inf 75.1%

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

   if -29500 < y.re < 1.34999999999999996e51

   1. Initial program 70.6%

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

   3. Taylor expanded in y.re around 0 60.8%

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

      1. associate-*r/ 60.8%

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

      2. neg-mul-1 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -29500 \lor \neg \left(y.re \leq 1.35 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+257} \lor \neg \left(y.im \leq 1.75 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.2e+257) (not (<= y.im 1.75e+172)))
   (/ x.im y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e+257) || !(y_46_im <= 1.75e+172)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-8.2d+257)) .or. (.not. (y_46im <= 1.75d+172))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e+257) || !(y_46_im <= 1.75e+172)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.2e+257) or not (y_46_im <= 1.75e+172):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.2e+257) || !(y_46_im <= 1.75e+172))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -8.2e+257) || ~((y_46_im <= 1.75e+172)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -8.2e+257], N[Not[LessEqual[y$46$im, 1.75e+172]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.20000000000000038e257 or 1.74999999999999989e172 < y.im

   1. Initial program 53.4%

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

      1. *-un-lft-identity 53.4%

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

      2. add-sqr-sqrt 53.4%

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

      3. times-frac 53.4%

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

      4. hypot-def 53.4%

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

      5. hypot-def 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in y.re around -inf 44.9%

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

      1. neg-mul-1 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in y.im around -inf 45.0%

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

   if -8.20000000000000038e257 < y.im < 1.74999999999999989e172

   1. Initial program 65.3%

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

   3. Taylor expanded in y.re around inf 49.3%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+257} \lor \neg \left(y.im \leq 1.75 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+169} \lor \neg \left(y.im \leq 5.1 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.4e+169) (not (<= y.im 5.1e+110)))
   (/ 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 <= -1.4e+169) || !(y_46_im <= 5.1e+110)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.4d+169)) .or. (.not. (y_46im <= 5.1d+110))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.4e+169) || !(y_46_im <= 5.1e+110)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.4e+169) or not (y_46_im <= 5.1e+110):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.4e+169) || !(y_46_im <= 5.1e+110))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.4e+169) || ~((y_46_im <= 5.1e+110)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.4e+169], N[Not[LessEqual[y$46$im, 5.1e+110]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.4000000000000001e169 or 5.1000000000000002e110 < y.im

   1. Initial program 50.9%

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

      1. *-un-lft-identity 50.9%

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

      2. add-sqr-sqrt 50.9%

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

      3. times-frac 50.8%

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

      4. hypot-def 50.8%

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

      5. hypot-def 65.3%

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

   4. Applied egg-rr 65.3%

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

   5. Taylor expanded in y.re around 0 71.8%

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

      1. neg-mul-1 71.8%

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

      2. +-commutative 71.8%

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

      3. unsub-neg 71.8%

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

      4. associate-/l* 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in y.im around -inf 45.3%

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

   if -1.4000000000000001e169 < y.im < 5.1000000000000002e110

   1. Initial program 67.5%

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

   3. Taylor expanded in y.re around inf 52.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+169} \lor \neg \left(y.im \leq 5.1 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 9.9% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_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_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 63.5%

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

   1. *-un-lft-identity 63.5%

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

   2. add-sqr-sqrt 63.5%

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

   3. times-frac 63.4%

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

   4. hypot-def 63.5%

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

   5. hypot-def 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in y.re around -inf 33.8%

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

   1. neg-mul-1 33.8%

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

7. Simplified 33.8%

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

8. Taylor expanded in y.im around -inf 14.9%

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

9. Final simplification 14.9%

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

Reproduce

herbie shell --seed 2024019 
(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))))