_divideComplex, imaginary part

Percentage Accurate: 61.2% → 99.3%

Time: 14.2s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.0× speedup

Math

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

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(Float64(y_46_re / hypot(y_46_re, y_46_im)) * x_46_im) - Float64(Float64(y_46_im / 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[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision] - N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. *-un-lft-identity 60.3%

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

   2. add-sqr-sqrt 60.3%

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

   3. times-frac 60.3%

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

   4. hypot-def 60.3%

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

   5. hypot-def 74.4%

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

3. Applied egg-rr 74.4%

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

   1. div-sub 74.4%

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

   2. *-commutative 74.4%

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

   3. *-commutative 74.4%

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

5. Applied egg-rr 74.4%

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

   1. associate-/l* 85.0%

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

   2. associate-/l* 97.9%

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

7. Simplified 97.9%

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

   1. associate-*l/ 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. associate-/r/ 98.9%

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

   4. associate-/r/ 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 88.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+229}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

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+229)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (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+229) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / 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+229)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / 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+229], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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))) < 9.9999999999999999e228

   1. Initial program 78.8%

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

      1. *-un-lft-identity 78.8%

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

      2. add-sqr-sqrt 78.8%

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

      3. times-frac 78.8%

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

      4. hypot-def 78.8%

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

      5. hypot-def 96.1%

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

   3. Applied egg-rr 96.1%

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

   if 9.9999999999999999e228 < (/.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 12.9%

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

      1. div-sub 9.8%

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

      2. sub-neg 9.8%

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

      3. *-commutative 9.8%

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

      4. add-sqr-sqrt 9.8%

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

      5. times-frac 13.7%

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

      6. fma-def 13.7%

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

      7. hypot-def 13.7%

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

      8. hypot-def 45.8%

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

      9. associate-/l* 50.2%

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

      10. add-sqr-sqrt 50.2%

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

      11. pow2 50.2%

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

      12. hypot-def 50.2%

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

   3. Applied egg-rr 50.2%

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

   4. Taylor expanded in y.re around 0 81.3%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+229}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternative 3: 87.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_1 (+ (* y.re y.re) (* y.im y.im))) 5e+257)
     (* t_0 (/ t_1 (hypot y.re y.im)))
     (* t_0 (- (/ y.re (/ (hypot y.re y.im) x.im)) x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+257) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+257) {
		tmp = t_0 * (t_1 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = t_0 * ((y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+257:
		tmp = t_0 * (t_1 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = t_0 * ((y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+257)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(t_0 * Float64(Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_1 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+257)
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	else
		tmp = t_0 * ((y_46_re / (hypot(y_46_re, y_46_im) / x_46_im)) - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+257], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.00000000000000028e257

   1. Initial program 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 79.0%

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

      4. hypot-def 79.0%

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

      5. hypot-def 96.1%

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

   3. Applied egg-rr 96.1%

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

   if 5.00000000000000028e257 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 10.4%

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

      1. *-un-lft-identity 10.4%

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

      2. add-sqr-sqrt 10.4%

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

      3. times-frac 10.4%

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

      4. hypot-def 10.4%

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

      5. hypot-def 16.7%

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

   3. Applied egg-rr 16.7%

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

      1. div-sub 16.7%

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

      2. *-commutative 16.7%

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

      3. *-commutative 16.7%

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

   5. Applied egg-rr 16.7%

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

      1. associate-/l* 54.2%

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

      2. associate-/l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y.im around inf 77.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{x.re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - x.re\right)\\ \end{array} \]

Alternative 4: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= y.re -2.7e+109)
     (* (/ 1.0 (hypot y.re y.im)) (- (/ x.re (/ y.re y.im)) x.im))
     (if (<= y.re -3.1e+46)
       (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im))
       (if (<= y.re -3.4e-22)
         (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 3.3e-94)
           (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
           (if (<= y.re 2.1e-5)
             (* t_0 (/ 1.0 (pow (hypot y.re y.im) 2.0)))
             (/
              (- 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) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_re <= -2.7e+109) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	} else if (y_46_re <= -3.1e+46) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_re <= -3.4e-22) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 3.3e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 2.1e-5) {
		tmp = t_0 * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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);
	double tmp;
	if (y_46_re <= -2.7e+109) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	} else if (y_46_re <= -3.1e+46) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_re <= -3.4e-22) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 3.3e-94) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 2.1e-5) {
		tmp = t_0 * (1.0 / Math.pow(Math.hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = (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);
	}
	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 y_46_re <= -2.7e+109:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im)
	elif y_46_re <= -3.1e+46:
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	elif y_46_re <= -3.4e-22:
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 3.3e-94:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_re <= 2.1e-5:
		tmp = t_0 * (1.0 / math.pow(math.hypot(y_46_re, y_46_im), 2.0))
	else:
		tmp = (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)
	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 (y_46_re <= -2.7e+109)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) - x_46_im));
	elseif (y_46_re <= -3.1e+46)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= -3.4e-22)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 3.3e-94)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_re <= 2.1e-5)
		tmp = Float64(t_0 * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = Float64(Float64(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
	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 (y_46_re <= -2.7e+109)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	elseif (y_46_re <= -3.1e+46)
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	elseif (y_46_re <= -3.4e-22)
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 3.3e-94)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_re <= 2.1e-5)
		tmp = t_0 * (1.0 / (hypot(y_46_re, y_46_im) ^ 2.0));
	else
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.7e+109], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.1e+46], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.4e-22], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.3e-94], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-5], N[(t$95$0 * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - 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. Split input into 6 regimes

2. if y.re < -2.70000000000000001e109

   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. Step-by-step derivation

      1. *-un-lft-identity 40.0%

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

      2. add-sqr-sqrt 40.0%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. hypot-def 54.7%

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

   3. Applied egg-rr 54.7%

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

   4. Taylor expanded in y.re around -inf 76.4%

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

      1. +-commutative 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unsub-neg 76.4%

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

      4. associate-/l* 85.9%

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

   6. Simplified 85.9%

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

   if -2.70000000000000001e109 < y.re < -3.09999999999999975e46

   1. Initial program 47.1%

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

   2. Taylor expanded in y.re around 0 77.8%

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

      1. +-commutative 77.8%

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

      2. mul-1-neg 77.8%

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

      3. unsub-neg 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/l* 77.8%

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

   4. Simplified 77.8%

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

      1. pow2 77.8%

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

      2. *-un-lft-identity 77.8%

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

      3. times-frac 77.8%

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

   6. Applied egg-rr 77.8%

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

   if -3.09999999999999975e46 < y.re < -3.3999999999999998e-22

   1. Initial program 73.7%

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

   if -3.3999999999999998e-22 < y.re < 3.3000000000000001e-94

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. add-sqr-sqrt 69.8%

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

      3. times-frac 69.8%

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

      4. hypot-def 69.8%

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

      5. hypot-def 82.5%

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

   3. Applied egg-rr 82.5%

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

   4. Taylor expanded in y.im around -inf 50.1%

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

   5. Taylor expanded in y.im around -inf 87.9%

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

   if 3.3000000000000001e-94 < y.re < 2.09999999999999988e-5

   1. Initial program 99.5%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. pow2 99.6%

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

      5. hypot-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   if 2.09999999999999988e-5 < y.re

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.4%

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

      4. hypot-def 49.4%

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

      5. hypot-def 69.8%

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

   3. Applied egg-rr 69.8%

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

      1. div-sub 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. associate-/l* 91.9%

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

      2. associate-/l* 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y.re around inf 88.8%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\right)\right)} \]

      2. expm1-udef 32.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\right)} - 1} \]

      3. associate-*l/ 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      5. associate-/r/ 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.im - \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      6. associate-*l/ 31.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.im - \color{blue}{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.im - \frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 70.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.im - \frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

       2. expm1-log1p 83.5%

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

       3. associate-/l* 88.9%

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

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\left(y.re \cdot x.im - y.im \cdot x.re\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.72 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

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 -2.8e+109)
     (* (/ 1.0 (hypot y.re y.im)) (- (/ x.re (/ y.re y.im)) x.im))
     (if (<= y.re -1.6e+47)
       (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im))
       (if (<= y.re -5.6e-23)
         t_0
         (if (<= y.re 6.8e-96)
           (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) y.im)))
           (if (<= y.re 1.72e-9)
             t_0
             (/
              (- 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) {
	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 <= -2.8e+109) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	} else if (y_46_re <= -1.6e+47) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.6e-23) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-96) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 1.72e-9) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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 <= -2.8e+109) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	} else if (y_46_re <= -1.6e+47) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_re <= -5.6e-23) {
		tmp = t_0;
	} else if (y_46_re <= 6.8e-96) {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_re <= 1.72e-9) {
		tmp = t_0;
	} else {
		tmp = (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);
	}
	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 <= -2.8e+109:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im)
	elif y_46_re <= -1.6e+47:
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im)
	elif y_46_re <= -5.6e-23:
		tmp = t_0
	elif y_46_re <= 6.8e-96:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_re <= 1.72e-9:
		tmp = t_0
	else:
		tmp = (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)
	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 <= -2.8e+109)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_re / Float64(y_46_re / y_46_im)) - x_46_im));
	elseif (y_46_re <= -1.6e+47)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_re <= -5.6e-23)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-96)
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_re <= 1.72e-9)
		tmp = t_0;
	else
		tmp = Float64(Float64(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
	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 <= -2.8e+109)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_re / (y_46_re / y_46_im)) - x_46_im);
	elseif (y_46_re <= -1.6e+47)
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	elseif (y_46_re <= -5.6e-23)
		tmp = t_0;
	elseif (y_46_re <= 6.8e-96)
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_re <= 1.72e-9)
		tmp = t_0;
	else
		tmp = (x_46_im - (y_46_im / (hypot(y_46_re, y_46_im) / x_46_re))) / 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, -2.8e+109], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.6e+47], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.6e-23], t$95$0, If[LessEqual[y$46$re, 6.8e-96], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.72e-9], t$95$0, N[(N[(x$46$im - 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. Split input into 5 regimes

2. if y.re < -2.8000000000000002e109

   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. Step-by-step derivation

      1. *-un-lft-identity 40.0%

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

      2. add-sqr-sqrt 40.0%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. hypot-def 54.7%

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

   3. Applied egg-rr 54.7%

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

   4. Taylor expanded in y.re around -inf 76.4%

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

      1. +-commutative 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unsub-neg 76.4%

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

      4. associate-/l* 85.9%

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

   6. Simplified 85.9%

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

   if -2.8000000000000002e109 < y.re < -1.6e47

   1. Initial program 47.1%

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

   2. Taylor expanded in y.re around 0 77.8%

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

      1. +-commutative 77.8%

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

      2. mul-1-neg 77.8%

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

      3. unsub-neg 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/l* 77.8%

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

   4. Simplified 77.8%

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

      1. pow2 77.8%

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

      2. *-un-lft-identity 77.8%

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

      3. times-frac 77.8%

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

   6. Applied egg-rr 77.8%

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

   if -1.6e47 < y.re < -5.5999999999999994e-23 or 6.8000000000000002e-96 < y.re < 1.72000000000000006e-9

   1. Initial program 87.8%

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

   if -5.5999999999999994e-23 < y.re < 6.8000000000000002e-96

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. add-sqr-sqrt 69.8%

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

      3. times-frac 69.8%

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

      4. hypot-def 69.8%

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

      5. hypot-def 82.5%

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

   3. Applied egg-rr 82.5%

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

   4. Taylor expanded in y.im around -inf 50.1%

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

   5. Taylor expanded in y.im around -inf 87.9%

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

   if 1.72000000000000006e-9 < y.re

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.4%

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

      4. hypot-def 49.4%

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

      5. hypot-def 69.8%

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

   3. Applied egg-rr 69.8%

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

      1. div-sub 69.8%

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

      2. *-commutative 69.8%

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

      3. *-commutative 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. associate-/l* 91.9%

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

      2. associate-/l* 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y.re around inf 88.8%

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

      1. expm1-log1p-u 75.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\right)\right)} \]

      2. expm1-udef 32.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)\right)} - 1} \]

      3. associate-*l/ 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      5. associate-/r/ 32.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.im - \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

      6. associate-*l/ 31.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x.im - \color{blue}{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   10. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.im - \frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 70.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.im - \frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

       2. expm1-log1p 83.5%

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

       3. associate-/l* 88.9%

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

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re}{\frac{y.re}{y.im}} - x.im\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.72 \cdot 10^{-9}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 78.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+72}:\\ \;\;\;\;t_1 \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \end{array} \]

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 (/ 1.0 (hypot y.re y.im))))
   (if (<= y.im -5e+72)
     (* t_1 (- x.re (* y.re (/ x.im y.im))))
     (if (<= y.im -7.8e-199)
       t_0
       (if (<= y.im 3.2e-128)
         (/ x.im y.re)
         (if (<= y.im 1.35e+73)
           t_0
           (* t_1 (- (/ x.im (/ y.im y.re)) x.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 = 1.0 / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -5e+72) {
		tmp = t_1 * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	} else if (y_46_im <= -7.8e-199) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.35e+73) {
		tmp = t_0;
	} else {
		tmp = t_1 * ((x_46_im / (y_46_im / y_46_re)) - x_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 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -5e+72) {
		tmp = t_1 * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	} else if (y_46_im <= -7.8e-199) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.35e+73) {
		tmp = t_0;
	} else {
		tmp = t_1 * ((x_46_im / (y_46_im / y_46_re)) - x_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 = 1.0 / math.hypot(y_46_re, y_46_im)
	tmp = 0
	if y_46_im <= -5e+72:
		tmp = t_1 * (x_46_re - (y_46_re * (x_46_im / y_46_im)))
	elif y_46_im <= -7.8e-199:
		tmp = t_0
	elif y_46_im <= 3.2e-128:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1.35e+73:
		tmp = t_0
	else:
		tmp = t_1 * ((x_46_im / (y_46_im / y_46_re)) - x_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(1.0 / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -5e+72)
		tmp = Float64(t_1 * Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im))));
	elseif (y_46_im <= -7.8e-199)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-128)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.35e+73)
		tmp = t_0;
	else
		tmp = Float64(t_1 * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_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 = 1.0 / hypot(y_46_re, y_46_im);
	tmp = 0.0;
	if (y_46_im <= -5e+72)
		tmp = t_1 * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	elseif (y_46_im <= -7.8e-199)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-128)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1.35e+73)
		tmp = t_0;
	else
		tmp = t_1 * ((x_46_im / (y_46_im / y_46_re)) - x_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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5e+72], N[(t$95$1 * N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.8e-199], t$95$0, If[LessEqual[y$46$im, 3.2e-128], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+73], t$95$0, N[(t$95$1 * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.99999999999999992e72

   1. Initial program 41.5%

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

      1. *-un-lft-identity 41.5%

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

      2. add-sqr-sqrt 41.5%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 63.4%

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

   3. Applied egg-rr 63.4%

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

   4. Taylor expanded in y.im around -inf 78.7%

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

      1. add-sqr-sqrt 42.9%

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

      2. fma-def 43.0%

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

      3. mul-1-neg 43.0%

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

      4. add-sqr-sqrt 25.2%

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

      5. sqrt-unprod 39.1%

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

      6. sqr-neg 39.1%

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

      7. mul-1-neg 39.1%

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

      8. mul-1-neg 39.1%

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

      9. sqrt-unprod 31.7%

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

      10. add-sqr-sqrt 38.2%

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

      11. fma-neg 38.2%

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

      12. add-sqr-sqrt 70.2%

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

      13. add-sqr-sqrt 50.9%

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

      14. sqrt-unprod 71.3%

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

      15. mul-1-neg 71.3%

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

      16. mul-1-neg 71.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/r/ 83.3%

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

   8. Simplified 83.3%

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

   if -4.99999999999999992e72 < y.im < -7.8000000000000002e-199 or 3.1999999999999998e-128 < y.im < 1.35e73

   1. Initial program 83.1%

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

   if -7.8000000000000002e-199 < y.im < 3.1999999999999998e-128

   1. Initial program 59.1%

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

   2. Taylor expanded in y.re around inf 78.8%

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

   if 1.35e73 < y.im

   1. Initial program 40.2%

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

      1. *-un-lft-identity 40.2%

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

      2. add-sqr-sqrt 40.2%

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

      3. times-frac 40.2%

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

      4. hypot-def 40.2%

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

      5. hypot-def 55.3%

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

   3. Applied egg-rr 55.3%

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

   4. Taylor expanded in y.re around 0 84.3%

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

      1. +-commutative 84.3%

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

      2. mul-1-neg 84.3%

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

      3. unsub-neg 84.3%

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

      4. associate-/l* 88.8%

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

   6. Simplified 88.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]

Alternative 7: 78.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.re - y.re \cdot \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_1\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot t_1\\ \end{array} \end{array} \]

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.re (* y.re (/ x.im y.im)))))
   (if (<= y.im -7e+71)
     (* (/ 1.0 (hypot y.re y.im)) t_1)
     (if (<= y.im -2.8e-198)
       t_0
       (if (<= y.im 7.6e-128)
         (/ x.im y.re)
         (if (<= y.im 5.4e+73) t_0 (* (/ -1.0 y.im) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 * (x_46_im / y_46_im));
	double tmp;
	if (y_46_im <= -7e+71) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_1;
	} else if (y_46_im <= -2.8e-198) {
		tmp = t_0;
	} else if (y_46_im <= 7.6e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 5.4e+73) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / y_46_im) * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 * (x_46_im / y_46_im));
	double tmp;
	if (y_46_im <= -7e+71) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * t_1;
	} else if (y_46_im <= -2.8e-198) {
		tmp = t_0;
	} else if (y_46_im <= 7.6e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 5.4e+73) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / y_46_im) * 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_re - (y_46_re * (x_46_im / y_46_im))
	tmp = 0
	if y_46_im <= -7e+71:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * t_1
	elif y_46_im <= -2.8e-198:
		tmp = t_0
	elif y_46_im <= 7.6e-128:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 5.4e+73:
		tmp = t_0
	else:
		tmp = (-1.0 / y_46_im) * 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(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -7e+71)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * t_1);
	elseif (y_46_im <= -2.8e-198)
		tmp = t_0;
	elseif (y_46_im <= 7.6e-128)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 5.4e+73)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((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 * (x_46_im / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -7e+71)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_1;
	elseif (y_46_im <= -2.8e-198)
		tmp = t_0;
	elseif (y_46_im <= 7.6e-128)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 5.4e+73)
		tmp = t_0;
	else
		tmp = (-1.0 / y_46_im) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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 * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7e+71], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$im, -2.8e-198], t$95$0, If[LessEqual[y$46$im, 7.6e-128], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.4e+73], t$95$0, N[(N[(-1.0 / y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.9999999999999998e71

   1. Initial program 41.5%

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

      1. *-un-lft-identity 41.5%

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

      2. add-sqr-sqrt 41.5%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 63.4%

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

   3. Applied egg-rr 63.4%

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

   4. Taylor expanded in y.im around -inf 78.7%

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

      1. add-sqr-sqrt 42.9%

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

      2. fma-def 43.0%

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

      3. mul-1-neg 43.0%

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

      4. add-sqr-sqrt 25.2%

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

      5. sqrt-unprod 39.1%

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

      6. sqr-neg 39.1%

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

      7. mul-1-neg 39.1%

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

      8. mul-1-neg 39.1%

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

      9. sqrt-unprod 31.7%

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

      10. add-sqr-sqrt 38.2%

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

      11. fma-neg 38.2%

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

      12. add-sqr-sqrt 70.2%

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

      13. add-sqr-sqrt 50.9%

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

      14. sqrt-unprod 71.3%

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

      15. mul-1-neg 71.3%

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

      16. mul-1-neg 71.3%

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

   6. Applied egg-rr 83.3%

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

      1. associate-/r/ 83.3%

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

   8. Simplified 83.3%

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

   if -6.9999999999999998e71 < y.im < -2.7999999999999999e-198 or 7.6000000000000005e-128 < y.im < 5.3999999999999998e73

   1. Initial program 83.1%

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

   if -2.7999999999999999e-198 < y.im < 7.6000000000000005e-128

   1. Initial program 59.1%

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

   2. Taylor expanded in y.re around inf 78.8%

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

   if 5.3999999999999998e73 < y.im

   1. Initial program 40.2%

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

      1. *-un-lft-identity 40.2%

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

      2. add-sqr-sqrt 40.2%

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

      3. times-frac 40.2%

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

      4. hypot-def 40.2%

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

      5. hypot-def 55.3%

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

   3. Applied egg-rr 55.3%

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

   4. Taylor expanded in y.im around -inf 27.1%

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

      1. add-sqr-sqrt 17.3%

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

      2. fma-def 17.3%

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

      3. mul-1-neg 17.3%

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

      4. add-sqr-sqrt 8.9%

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

      5. sqrt-unprod 19.2%

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

      6. sqr-neg 19.2%

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

      7. mul-1-neg 19.2%

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

      8. mul-1-neg 19.2%

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

      9. sqrt-unprod 18.8%

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

      10. add-sqr-sqrt 21.3%

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

      11. fma-neg 21.3%

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

      12. add-sqr-sqrt 34.7%

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

      13. add-sqr-sqrt 28.4%

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

      14. sqrt-unprod 31.0%

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

      15. mul-1-neg 31.0%

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

      16. mul-1-neg 31.0%

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

   6. Applied egg-rr 27.2%

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

      1. associate-/r/ 27.2%

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

   8. Simplified 27.2%

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

   9. Taylor expanded in y.im around -inf 88.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 8: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (* (/ -1.0 y.im) (- x.re (* y.re (/ x.im y.im))))))
   (if (<= y.im -4e+84)
     t_1
     (if (<= y.im -9.2e-198)
       t_0
       (if (<= y.im 5.3e-128) (/ x.im y.re) (if (<= y.im 1e+73) 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 = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	double tmp;
	if (y_46_im <= -4e+84) {
		tmp = t_1;
	} else if (y_46_im <= -9.2e-198) {
		tmp = t_0;
	} else if (y_46_im <= 5.3e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1e+73) {
		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 = ((-1.0d0) / y_46im) * (x_46re - (y_46re * (x_46im / y_46im)))
    if (y_46im <= (-4d+84)) then
        tmp = t_1
    else if (y_46im <= (-9.2d-198)) then
        tmp = t_0
    else if (y_46im <= 5.3d-128) then
        tmp = x_46im / y_46re
    else if (y_46im <= 1d+73) 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 = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	double tmp;
	if (y_46_im <= -4e+84) {
		tmp = t_1;
	} else if (y_46_im <= -9.2e-198) {
		tmp = t_0;
	} else if (y_46_im <= 5.3e-128) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1e+73) {
		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 = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)))
	tmp = 0
	if y_46_im <= -4e+84:
		tmp = t_1
	elif y_46_im <= -9.2e-198:
		tmp = t_0
	elif y_46_im <= 5.3e-128:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1e+73:
		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(-1.0 / y_46_im) * Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im))))
	tmp = 0.0
	if (y_46_im <= -4e+84)
		tmp = t_1;
	elseif (y_46_im <= -9.2e-198)
		tmp = t_0;
	elseif (y_46_im <= 5.3e-128)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1e+73)
		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 = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	tmp = 0.0;
	if (y_46_im <= -4e+84)
		tmp = t_1;
	elseif (y_46_im <= -9.2e-198)
		tmp = t_0;
	elseif (y_46_im <= 5.3e-128)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1e+73)
		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[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+84], t$95$1, If[LessEqual[y$46$im, -9.2e-198], t$95$0, If[LessEqual[y$46$im, 5.3e-128], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1e+73], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.00000000000000023e84 or 9.99999999999999983e72 < y.im

   1. Initial program 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. times-frac 38.6%

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

      4. hypot-def 38.6%

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

      5. hypot-def 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in y.im around -inf 52.5%

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

      1. add-sqr-sqrt 30.1%

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

      2. fma-def 30.1%

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

      3. mul-1-neg 30.1%

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

      4. add-sqr-sqrt 17.2%

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

      5. sqrt-unprod 28.9%

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

      6. sqr-neg 28.9%

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

      7. mul-1-neg 28.9%

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

      8. mul-1-neg 28.9%

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

      9. sqrt-unprod 25.7%

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

      10. add-sqr-sqrt 29.4%

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

      11. fma-neg 29.4%

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

      12. add-sqr-sqrt 52.6%

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

      13. add-sqr-sqrt 39.9%

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

      14. sqrt-unprod 51.3%

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

      15. mul-1-neg 51.3%

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

      16. mul-1-neg 51.3%

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

   6. Applied egg-rr 55.2%

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

      1. associate-/r/ 55.2%

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

   8. Simplified 55.2%

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

   9. Taylor expanded in y.im around -inf 85.1%

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

   if -4.00000000000000023e84 < y.im < -9.20000000000000053e-198 or 5.2999999999999999e-128 < y.im < 9.99999999999999983e72

   1. Initial program 83.8%

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

   if -9.20000000000000053e-198 < y.im < 5.2999999999999999e-128

   1. Initial program 59.1%

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

   2. Taylor expanded in y.re around inf 78.8%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+84}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+73}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 9: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+114} \lor \neg \left(y.re \leq 2.2 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.75e+114) (not (<= y.re 2.2e+26)))
   (/ x.im y.re)
   (* (/ -1.0 y.im) (- x.re (* y.re (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.75e+114) || !(y_46_re <= 2.2e+26)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.75d+114)) .or. (.not. (y_46re <= 2.2d+26))) then
        tmp = x_46im / y_46re
    else
        tmp = ((-1.0d0) / y_46im) * (x_46re - (y_46re * (x_46im / y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.75e+114) || !(y_46_re <= 2.2e+26)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.75e+114) or not (y_46_re <= 2.2e+26):
		tmp = x_46_im / y_46_re
	else:
		tmp = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.75e+114) || !(y_46_re <= 2.2e+26))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.75e+114) || ~((y_46_re <= 2.2e+26)))
		tmp = x_46_im / y_46_re;
	else
		tmp = (-1.0 / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.75e+114], N[Not[LessEqual[y$46$re, 2.2e+26]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.75e114 or 2.20000000000000007e26 < y.re

   1. Initial program 44.8%

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

   2. Taylor expanded in y.re around inf 68.9%

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

   if -1.75e114 < y.re < 2.20000000000000007e26

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.7%

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

      4. hypot-def 71.7%

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

      5. hypot-def 82.6%

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

   3. Applied egg-rr 82.6%

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

   4. Taylor expanded in y.im around -inf 44.0%

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

      1. add-sqr-sqrt 18.4%

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

      2. fma-def 18.4%

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

      3. mul-1-neg 18.4%

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

      4. add-sqr-sqrt 11.3%

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

      5. sqrt-unprod 19.7%

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

      6. sqr-neg 19.7%

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

      7. mul-1-neg 19.7%

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

      8. mul-1-neg 19.7%

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

      9. sqrt-unprod 18.0%

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

      10. add-sqr-sqrt 23.5%

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

      11. fma-neg 23.5%

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

      12. add-sqr-sqrt 46.5%

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

      13. add-sqr-sqrt 33.2%

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

      14. sqrt-unprod 43.3%

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

      15. mul-1-neg 43.3%

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

      16. mul-1-neg 43.3%

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

   6. Applied egg-rr 44.6%

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

      1. associate-/r/ 44.7%

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

   8. Simplified 44.7%

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

   9. Taylor expanded in y.im around -inf 78.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+114} \lor \neg \left(y.re \leq 2.2 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 10: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+109} \lor \neg \left(y.re \leq 3.1 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.2e+109) (not (<= y.re 3.1e+27)))
   (/ x.im y.re)
   (* (/ -1.0 y.im) (- x.re (/ (* y.re x.im) 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 <= -3.2e+109) || !(y_46_re <= 3.1e+27)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / 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 <= (-3.2d+109)) .or. (.not. (y_46re <= 3.1d+27))) then
        tmp = x_46im / y_46re
    else
        tmp = ((-1.0d0) / y_46im) * (x_46re - ((y_46re * x_46im) / 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 <= -3.2e+109) || !(y_46_re <= 3.1e+27)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / 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 <= -3.2e+109) or not (y_46_re <= 3.1e+27):
		tmp = x_46_im / y_46_re
	else:
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / 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 <= -3.2e+109) || !(y_46_re <= 3.1e+27))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-1.0 / y_46_im) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / 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 <= -3.2e+109) || ~((y_46_re <= 3.1e+27)))
		tmp = x_46_im / y_46_re;
	else
		tmp = (-1.0 / y_46_im) * (x_46_re - ((y_46_re * x_46_im) / 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, -3.2e+109], N[Not[LessEqual[y$46$re, 3.1e+27]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(-1.0 / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.2000000000000001e109 or 3.09999999999999996e27 < y.re

   1. Initial program 44.8%

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

   2. Taylor expanded in y.re around inf 68.9%

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

   if -3.2000000000000001e109 < y.re < 3.09999999999999996e27

   1. Initial program 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. add-sqr-sqrt 71.8%

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

      3. times-frac 71.7%

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

      4. hypot-def 71.7%

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

      5. hypot-def 82.6%

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

   3. Applied egg-rr 82.6%

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

   4. Taylor expanded in y.im around -inf 44.0%

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

   5. Taylor expanded in y.im around -inf 79.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+109} \lor \neg \left(y.re \leq 3.1 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \end{array} \]

Alternative 11: 61.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-67} \lor \neg \left(y.im \leq 2.4 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.5e-67) (not (<= y.im 2.4e+88)))
   (/ (- 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 <= -7.5e-67) || !(y_46_im <= 2.4e+88)) {
		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 <= (-7.5d-67)) .or. (.not. (y_46im <= 2.4d+88))) 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 <= -7.5e-67) || !(y_46_im <= 2.4e+88)) {
		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 <= -7.5e-67) or not (y_46_im <= 2.4e+88):
		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 <= -7.5e-67) || !(y_46_im <= 2.4e+88))
		tmp = Float64(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 <= -7.5e-67) || ~((y_46_im <= 2.4e+88)))
		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, -7.5e-67], N[Not[LessEqual[y$46$im, 2.4e+88]], $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 < -7.5000000000000005e-67 or 2.3999999999999999e88 < y.im

   1. Initial program 49.9%

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

   2. Taylor expanded in y.re around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

   if -7.5000000000000005e-67 < y.im < 2.3999999999999999e88

   1. Initial program 70.8%

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

   2. Taylor expanded in y.re around inf 63.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-67} \lor \neg \left(y.im \leq 2.4 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12: 45.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+264} \lor \neg \left(y.im \leq 4.4 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.7e+264) (not (<= y.im 4.4e+152)))
   (/ x.re y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.7e+264) || !(y_46_im <= 4.4e+152)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.7d+264)) .or. (.not. (y_46im <= 4.4d+152))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.7e+264) || !(y_46_im <= 4.4e+152)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.7e+264) or not (y_46_im <= 4.4e+152):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.7e+264) || !(y_46_im <= 4.4e+152))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.7e+264) || ~((y_46_im <= 4.4e+152)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.7e+264], N[Not[LessEqual[y$46$im, 4.4e+152]], $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 < -1.7e264 or 4.3999999999999996e152 < y.im

   1. Initial program 36.5%

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

      1. *-un-lft-identity 36.5%

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

      2. add-sqr-sqrt 36.5%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 52.3%

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

   3. Applied egg-rr 52.3%

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

   4. Taylor expanded in y.im around -inf 49.6%

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

   5. Taylor expanded in y.re around 0 35.7%

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

   if -1.7e264 < y.im < 4.3999999999999996e152

   1. Initial program 66.2%

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

   2. Taylor expanded in y.re around inf 49.6%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+264} \lor \neg \left(y.im \leq 4.4 \cdot 10^{+152}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 13: 43.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 2.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 2.9e+151) (/ x.im y.re) (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.9e+151) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 2.9d+151) then
        tmp = x_46im / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.9e+151) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 2.9e+151:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 2.9e+151)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 2.9e+151)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 2.9e+151], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 2.90000000000000018e151

   1. Initial program 64.6%

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

   2. Taylor expanded in y.re around inf 47.0%

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

   if 2.90000000000000018e151 < y.im

   1. Initial program 35.0%

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

      1. *-un-lft-identity 35.0%

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

      2. add-sqr-sqrt 35.0%

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

      3. times-frac 34.9%

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

      4. hypot-def 34.9%

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

      5. hypot-def 52.6%

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

   3. Applied egg-rr 52.6%

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

   4. Taylor expanded in y.im around -inf 33.0%

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

   5. Taylor expanded in y.re around -inf 22.6%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 2.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 14: 9.4% 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 60.3%

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

   1. *-un-lft-identity 60.3%

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

   2. add-sqr-sqrt 60.3%

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

   3. times-frac 60.3%

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

   4. hypot-def 60.3%

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

   5. hypot-def 74.4%

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

3. Applied egg-rr 74.4%

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

4. Taylor expanded in y.im around -inf 33.6%

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

5. Taylor expanded in y.re around -inf 11.8%

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

6. Final simplification 11.8%

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

Reproduce

herbie shell --seed 2023311 
(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))))