_divideComplex, real part

Percentage Accurate: 61.5% → 84.2%

Time: 11.1s

Alternatives: 11

Speedup: 2.1×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 84.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+234)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+234) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+234)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+234], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000003e234

   1. Initial program 79.5%

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

      1. add-sqr-sqrt 79.5%

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

      2. *-un-lft-identity 79.5%

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

      3. times-frac 79.5%

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

      4. hypot-def 79.5%

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

      5. fma-def 79.5%

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

      6. hypot-def 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 5.0000000000000003e234 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 17.8%

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

      1. add-sqr-sqrt 17.8%

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

      2. *-un-lft-identity 17.8%

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

      3. times-frac 17.8%

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

      4. hypot-def 17.8%

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

      5. fma-def 17.8%

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

      6. hypot-def 23.5%

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

   4. Applied egg-rr 23.5%

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

   5. Taylor expanded in y.re around 0 56.8%

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

      1. associate-/l* 57.3%

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

   7. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. *-un-lft-identity 57.3%

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

      3. times-frac 68.4%

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

   9. Applied egg-rr 68.4%

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

       1. /-rgt-identity 68.4%

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

       2. *-un-lft-identity 68.4%

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

       3. times-frac 70.3%

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

   11. Applied egg-rr 70.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 5e+234)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ x.re (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+234) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+234) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+234:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+234)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+234)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $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], 5e+234], 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[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000003e234

   1. Initial program 79.5%

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

      1. add-sqr-sqrt 79.5%

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

      2. *-un-lft-identity 79.5%

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

      3. times-frac 79.5%

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

      4. hypot-def 79.5%

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

      5. fma-def 79.5%

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

      6. hypot-def 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. fma-def 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 5.0000000000000003e234 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 17.8%

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

      1. add-sqr-sqrt 17.8%

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

      2. *-un-lft-identity 17.8%

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

      3. times-frac 17.8%

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

      4. hypot-def 17.8%

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

      5. fma-def 17.8%

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

      6. hypot-def 23.5%

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

   4. Applied egg-rr 23.5%

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

   5. Taylor expanded in y.re around 0 56.8%

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

      1. associate-/l* 57.3%

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

   7. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. *-un-lft-identity 57.3%

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

      3. times-frac 68.4%

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

   9. Applied egg-rr 68.4%

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

       1. /-rgt-identity 68.4%

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

       2. *-un-lft-identity 68.4%

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

       3. times-frac 70.3%

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

   11. Applied egg-rr 70.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot t_1\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ x.re (/ y.im y.re))))
   (if (<= y.im -7.6e+104)
     (+ (/ x.im y.im) (* (/ 1.0 y.im) t_1))
     (if (<= y.im -2.4e-133)
       t_0
       (if (<= y.im 3.95e-115)
         (* (/ -1.0 y.re) (- (/ (* y.im (- x.im)) y.re) x.re))
         (if (<= y.im 2.8e+95)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.im t_1))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -7.6e+104) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * t_1);
	} else if (y_46_im <= -2.4e-133) {
		tmp = t_0;
	} else if (y_46_im <= 3.95e-115) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else if (y_46_im <= 2.8e+95) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + t_1);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -7.6e+104) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * t_1);
	} else if (y_46_im <= -2.4e-133) {
		tmp = t_0;
	} else if (y_46_im <= 3.95e-115) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else if (y_46_im <= 2.8e+95) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im + t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -7.6e+104:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * t_1)
	elif y_46_im <= -2.4e-133:
		tmp = t_0
	elif y_46_im <= 3.95e-115:
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re)
	elif y_46_im <= 2.8e+95:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im + t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -7.6e+104)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * t_1));
	elseif (y_46_im <= -2.4e-133)
		tmp = t_0;
	elseif (y_46_im <= 3.95e-115)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(Float64(y_46_im * Float64(-x_46_im)) / y_46_re) - x_46_re));
	elseif (y_46_im <= 2.8e+95)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_re / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -7.6e+104)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * t_1);
	elseif (y_46_im <= -2.4e-133)
		tmp = t_0;
	elseif (y_46_im <= 3.95e-115)
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	elseif (y_46_im <= 2.8e+95)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.6e+104], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.4e-133], t$95$0, If[LessEqual[y$46$im, 3.95e-115], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(N[(y$46$im * (-x$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+95], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.59999999999999938e104

   1. Initial program 40.0%

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

      1. add-sqr-sqrt 40.0%

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

      2. *-un-lft-identity 40.0%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. fma-def 40.0%

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

      6. hypot-def 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.2%

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

   7. Simplified 74.2%

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

      1. unpow2 74.2%

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

      2. *-un-lft-identity 74.2%

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

      3. times-frac 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. /-rgt-identity 83.1%

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

       2. *-un-lft-identity 83.1%

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

       3. times-frac 85.3%

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

   11. Applied egg-rr 85.3%

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

   if -7.59999999999999938e104 < y.im < -2.4e-133 or 3.95000000000000002e-115 < y.im < 2.7999999999999998e95

   1. Initial program 91.1%

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

   if -2.4e-133 < y.im < 3.95000000000000002e-115

   1. Initial program 75.8%

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

      1. add-sqr-sqrt 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 75.8%

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

      4. hypot-def 75.8%

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

      5. fma-def 75.8%

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

      6. hypot-def 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y.re around -inf 45.9%

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

      1. neg-mul-1 45.9%

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

      2. +-commutative 45.9%

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

      3. unsub-neg 45.9%

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

      4. mul-1-neg 45.9%

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

      5. distribute-neg-frac 45.9%

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

      6. distribute-rgt-neg-out 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in y.re around -inf 94.5%

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

   if 2.7999999999999998e95 < y.im

   1. Initial program 33.0%

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

      1. add-sqr-sqrt 33.0%

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

      2. *-un-lft-identity 33.0%

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

      3. times-frac 33.0%

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

      4. hypot-def 33.0%

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

      5. fma-def 33.0%

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

      6. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in y.re around 0 88.0%

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

      1. associate-/l* 94.0%

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

   7. Simplified 94.0%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.95 \cdot 10^{-115}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot t_1\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{t_0}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))) (t_1 (/ x.re (/ y.im y.re))))
   (if (<= y.im -2.35e+104)
     (+ (/ x.im y.im) (* (/ 1.0 y.im) t_1))
     (if (<= y.im -1.05e-129)
       (/ (fma y.re x.re (* x.im y.im)) t_0)
       (if (<= y.im 1.6e-114)
         (* (/ -1.0 y.re) (- (/ (* y.im (- x.im)) y.re) x.re))
         (if (<= y.im 3e+92)
           (/ (+ (* x.re y.re) (* x.im y.im)) t_0)
           (* (/ 1.0 (hypot y.re y.im)) (+ x.im t_1))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -2.35e+104) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * t_1);
	} else if (y_46_im <= -1.05e-129) {
		tmp = fma(y_46_re, x_46_re, (x_46_im * y_46_im)) / t_0;
	} else if (y_46_im <= 1.6e-114) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else if (y_46_im <= 3e+92) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + t_1);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -2.35e+104)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * t_1));
	elseif (y_46_im <= -1.05e-129)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(x_46_im * y_46_im)) / t_0);
	elseif (y_46_im <= 1.6e-114)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(Float64(y_46_im * Float64(-x_46_im)) / y_46_re) - x_46_re));
	elseif (y_46_im <= 3e+92)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / t_0);
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + t_1));
	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 * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.35e+104], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.05e-129], N[(N[(y$46$re * x$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.6e-114], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(N[(y$46$im * (-x$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3e+92], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.35000000000000008e104

   1. Initial program 40.0%

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

      1. add-sqr-sqrt 40.0%

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

      2. *-un-lft-identity 40.0%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. fma-def 40.0%

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

      6. hypot-def 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.2%

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

   7. Simplified 74.2%

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

      1. unpow2 74.2%

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

      2. *-un-lft-identity 74.2%

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

      3. times-frac 83.1%

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

   9. Applied egg-rr 83.1%

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

       1. /-rgt-identity 83.1%

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

       2. *-un-lft-identity 83.1%

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

       3. times-frac 85.3%

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

   11. Applied egg-rr 85.3%

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

   if -2.35000000000000008e104 < y.im < -1.05e-129

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. fma-def 87.2%

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

   4. Applied egg-rr 87.2%

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

   if -1.05e-129 < y.im < 1.6000000000000001e-114

   1. Initial program 75.8%

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

      1. add-sqr-sqrt 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 75.8%

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

      4. hypot-def 75.8%

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

      5. fma-def 75.8%

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

      6. hypot-def 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y.re around -inf 45.9%

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

      1. neg-mul-1 45.9%

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

      2. +-commutative 45.9%

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

      3. unsub-neg 45.9%

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

      4. mul-1-neg 45.9%

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

      5. distribute-neg-frac 45.9%

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

      6. distribute-rgt-neg-out 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in y.re around -inf 94.5%

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

   if 1.6000000000000001e-114 < y.im < 3.00000000000000013e92

   1. Initial program 97.0%

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

   if 3.00000000000000013e92 < y.im

   1. Initial program 33.0%

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

      1. add-sqr-sqrt 33.0%

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

      2. *-un-lft-identity 33.0%

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

      3. times-frac 33.0%

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

      4. hypot-def 33.0%

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

      5. fma-def 33.0%

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

      6. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in y.re around 0 88.0%

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

      1. associate-/l* 94.0%

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

   7. Simplified 94.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ x.re (/ y.im y.re))))))
   (if (<= y.im -1.6e+106)
     t_1
     (if (<= y.im -3.2e-135)
       t_0
       (if (<= y.im 6.4e-115)
         (* (/ -1.0 y.re) (- (/ (* y.im (- x.im)) y.re) x.re))
         (if (<= y.im 7.2e+91) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -1.6e+106) {
		tmp = t_1;
	} else if (y_46_im <= -3.2e-135) {
		tmp = t_0;
	} else if (y_46_im <= 6.4e-115) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else if (y_46_im <= 7.2e+91) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46im) + ((1.0d0 / y_46im) * (x_46re / (y_46im / y_46re)))
    if (y_46im <= (-1.6d+106)) then
        tmp = t_1
    else if (y_46im <= (-3.2d-135)) then
        tmp = t_0
    else if (y_46im <= 6.4d-115) then
        tmp = ((-1.0d0) / y_46re) * (((y_46im * -x_46im) / y_46re) - x_46re)
    else if (y_46im <= 7.2d+91) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -1.6e+106) {
		tmp = t_1;
	} else if (y_46_im <= -3.2e-135) {
		tmp = t_0;
	} else if (y_46_im <= 6.4e-115) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else if (y_46_im <= 7.2e+91) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_im <= -1.6e+106:
		tmp = t_1
	elif y_46_im <= -3.2e-135:
		tmp = t_0
	elif y_46_im <= 6.4e-115:
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re)
	elif y_46_im <= 7.2e+91:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_im <= -1.6e+106)
		tmp = t_1;
	elseif (y_46_im <= -3.2e-135)
		tmp = t_0;
	elseif (y_46_im <= 6.4e-115)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(Float64(y_46_im * Float64(-x_46_im)) / y_46_re) - x_46_re));
	elseif (y_46_im <= 7.2e+91)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_im <= -1.6e+106)
		tmp = t_1;
	elseif (y_46_im <= -3.2e-135)
		tmp = t_0;
	elseif (y_46_im <= 6.4e-115)
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	elseif (y_46_im <= 7.2e+91)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+106], t$95$1, If[LessEqual[y$46$im, -3.2e-135], t$95$0, If[LessEqual[y$46$im, 6.4e-115], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(N[(y$46$im * (-x$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+91], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.5999999999999999e106 or 7.2e91 < y.im

   1. Initial program 36.3%

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

      1. add-sqr-sqrt 36.3%

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

      2. *-un-lft-identity 36.3%

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

      3. times-frac 36.3%

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

      4. hypot-def 36.3%

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

      5. fma-def 36.3%

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

      6. hypot-def 62.0%

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

   4. Applied egg-rr 62.0%

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

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

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

      1. associate-/l* 77.7%

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

   7. Simplified 77.7%

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

      1. unpow2 77.7%

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

      2. *-un-lft-identity 77.7%

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

      3. times-frac 84.8%

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

   9. Applied egg-rr 84.8%

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

       1. /-rgt-identity 84.8%

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

       2. *-un-lft-identity 84.8%

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

       3. times-frac 89.8%

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

   11. Applied egg-rr 89.8%

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

   if -1.5999999999999999e106 < y.im < -3.2e-135 or 6.4e-115 < y.im < 7.2e91

   1. Initial program 91.1%

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

   if -3.2e-135 < y.im < 6.4e-115

   1. Initial program 75.8%

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

      1. add-sqr-sqrt 75.8%

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

      2. *-un-lft-identity 75.8%

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

      3. times-frac 75.8%

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

      4. hypot-def 75.8%

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

      5. fma-def 75.8%

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

      6. hypot-def 91.8%

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

   4. Applied egg-rr 91.8%

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

   5. Taylor expanded in y.re around -inf 45.9%

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

      1. neg-mul-1 45.9%

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

      2. +-commutative 45.9%

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

      3. unsub-neg 45.9%

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

      4. mul-1-neg 45.9%

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

      5. distribute-neg-frac 45.9%

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

      6. distribute-rgt-neg-out 45.9%

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

   7. Simplified 45.9%

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

   8. Taylor expanded in y.re around -inf 94.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -2.6e+92)
     t_0
     (if (<= y.im -1.05e+32)
       (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im -6.8e-77)
         (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
         (if (<= y.im 5.5e-54)
           (* (/ -1.0 y.re) (- (/ (* y.im (- x.im)) y.re) x.re))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -2.6e+92) {
		tmp = t_0;
	} else if (y_46_im <= -1.05e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -6.8e-77) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 5.5e-54) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46im <= (-2.6d+92)) then
        tmp = t_0
    else if (y_46im <= (-1.05d+32)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= (-6.8d-77)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= 5.5d-54) then
        tmp = ((-1.0d0) / y_46re) * (((y_46im * -x_46im) / y_46re) - x_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -2.6e+92) {
		tmp = t_0;
	} else if (y_46_im <= -1.05e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -6.8e-77) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 5.5e-54) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -2.6e+92:
		tmp = t_0
	elif y_46_im <= -1.05e+32:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= -6.8e-77:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= 5.5e-54:
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.6e+92)
		tmp = t_0;
	elseif (y_46_im <= -1.05e+32)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= -6.8e-77)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= 5.5e-54)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(Float64(y_46_im * Float64(-x_46_im)) / y_46_re) - x_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.6e+92)
		tmp = t_0;
	elseif (y_46_im <= -1.05e+32)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= -6.8e-77)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= 5.5e-54)
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.6e+92], t$95$0, If[LessEqual[y$46$im, -1.05e+32], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6.8e-77], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.5e-54], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(N[(y$46$im * (-x$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.5999999999999999e92 or 5.50000000000000046e-54 < y.im

   1. Initial program 49.4%

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

      1. add-sqr-sqrt 49.4%

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

      2. *-un-lft-identity 49.4%

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

      3. times-frac 49.4%

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

      4. hypot-def 49.4%

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

      5. fma-def 49.4%

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

      6. hypot-def 70.2%

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

   4. Applied egg-rr 70.2%

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

   5. Taylor expanded in y.re around 0 74.4%

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

      1. associate-/l* 73.0%

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

   7. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. *-un-lft-identity 73.0%

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

      3. times-frac 78.4%

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

   9. Applied egg-rr 78.4%

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

       1. /-rgt-identity 78.4%

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

       2. *-un-lft-identity 78.4%

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

       3. *-commutative 78.4%

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

       4. times-frac 84.0%

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

       5. clear-num 83.9%

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

   11. Applied egg-rr 83.9%

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

   if -2.5999999999999999e92 < y.im < -1.05e32

   1. Initial program 87.1%

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

   3. Taylor expanded in x.re around 0 87.1%

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

   if -1.05e32 < y.im < -6.79999999999999966e-77

   1. Initial program 84.5%

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

      1. add-sqr-sqrt 84.5%

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

      2. *-un-lft-identity 84.5%

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

      3. times-frac 84.4%

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

      4. hypot-def 84.4%

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

      5. fma-def 84.4%

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

      6. hypot-def 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in y.re around 0 65.3%

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

      1. associate-/l* 65.2%

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

   7. Simplified 65.2%

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

      1. unpow2 65.2%

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

      2. *-un-lft-identity 65.2%

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

      3. times-frac 65.2%

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

   9. Applied egg-rr 65.2%

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

       1. clear-num 65.2%

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

       2. associate-/r/ 65.2%

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

       3. /-rgt-identity 65.2%

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

       4. *-commutative 65.2%

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

       5. associate-/r* 65.2%

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

       6. clear-num 65.3%

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

   11. Applied egg-rr 65.3%

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

   if -6.79999999999999966e-77 < y.im < 5.50000000000000046e-54

   1. Initial program 80.5%

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

      1. add-sqr-sqrt 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. times-frac 80.5%

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

      4. hypot-def 80.5%

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

      5. fma-def 80.5%

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

      6. hypot-def 93.7%

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y.re around -inf 42.6%

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

      1. neg-mul-1 42.6%

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

      2. +-commutative 42.6%

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

      3. unsub-neg 42.6%

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

      4. mul-1-neg 42.6%

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

      5. distribute-neg-frac 42.6%

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

      6. distribute-rgt-neg-out 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y.re around -inf 90.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.55 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.8e+92)
   (+ (/ x.im y.im) (* (/ 1.0 y.im) (/ x.re (/ y.im y.re))))
   (if (<= y.im -4.4e+32)
     (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im -2.55e-76)
       (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
       (if (<= y.im 7.6e-53)
         (* (/ -1.0 y.re) (- (/ (* y.im (- x.im)) y.re) x.re))
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+92) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_im <= -4.4e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -2.55e-76) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 7.6e-53) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.8d+92)) then
        tmp = (x_46im / y_46im) + ((1.0d0 / y_46im) * (x_46re / (y_46im / y_46re)))
    else if (y_46im <= (-4.4d+32)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= (-2.55d-76)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= 7.6d-53) then
        tmp = ((-1.0d0) / y_46re) * (((y_46im * -x_46im) / y_46re) - x_46re)
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+92) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_im <= -4.4e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -2.55e-76) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 7.6e-53) {
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.8e+92:
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)))
	elif y_46_im <= -4.4e+32:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= -2.55e-76:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= 7.6e-53:
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re)
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.8e+92)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(1.0 / y_46_im) * Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= -4.4e+32)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= -2.55e-76)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= 7.6e-53)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(Float64(y_46_im * Float64(-x_46_im)) / y_46_re) - x_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.8e+92)
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_im <= -4.4e+32)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= -2.55e-76)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= 7.6e-53)
		tmp = (-1.0 / y_46_re) * (((y_46_im * -x_46_im) / y_46_re) - x_46_re);
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.8e+92], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.4e+32], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.55e-76], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.6e-53], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(N[(y$46$im * (-x$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.80000000000000001e92

   1. Initial program 46.0%

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

      1. add-sqr-sqrt 46.0%

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

      2. *-un-lft-identity 46.0%

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

      3. times-frac 46.0%

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

      4. hypot-def 46.0%

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

      5. fma-def 46.0%

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

      6. hypot-def 66.3%

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

   4. Applied egg-rr 66.3%

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

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

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

      1. associate-/l* 73.0%

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

   7. Simplified 73.0%

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

      1. unpow2 73.0%

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

      2. *-un-lft-identity 73.0%

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

      3. times-frac 81.0%

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

   9. Applied egg-rr 81.0%

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

       1. /-rgt-identity 81.0%

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

       2. *-un-lft-identity 81.0%

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

       3. times-frac 86.7%

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

   11. Applied egg-rr 86.7%

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

   if -2.80000000000000001e92 < y.im < -4.40000000000000002e32

   1. Initial program 87.1%

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

   3. Taylor expanded in x.re around 0 87.1%

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

   if -4.40000000000000002e32 < y.im < -2.54999999999999993e-76

   1. Initial program 84.5%

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

      1. add-sqr-sqrt 84.5%

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

      2. *-un-lft-identity 84.5%

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

      3. times-frac 84.4%

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

      4. hypot-def 84.4%

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

      5. fma-def 84.4%

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

      6. hypot-def 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in y.re around 0 65.3%

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

      1. associate-/l* 65.2%

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

   7. Simplified 65.2%

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

      1. unpow2 65.2%

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

      2. *-un-lft-identity 65.2%

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

      3. times-frac 65.2%

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

   9. Applied egg-rr 65.2%

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

       1. clear-num 65.2%

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

       2. associate-/r/ 65.2%

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

       3. /-rgt-identity 65.2%

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

       4. *-commutative 65.2%

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

       5. associate-/r* 65.2%

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

       6. clear-num 65.3%

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

   11. Applied egg-rr 65.3%

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

   if -2.54999999999999993e-76 < y.im < 7.5999999999999995e-53

   1. Initial program 80.5%

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

      1. add-sqr-sqrt 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. times-frac 80.5%

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

      4. hypot-def 80.5%

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

      5. fma-def 80.5%

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

      6. hypot-def 93.7%

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y.re around -inf 42.6%

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

      1. neg-mul-1 42.6%

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

      2. +-commutative 42.6%

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

      3. unsub-neg 42.6%

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

      4. mul-1-neg 42.6%

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

      5. distribute-neg-frac 42.6%

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

      6. distribute-rgt-neg-out 42.6%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y.re around -inf 90.0%

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

   if 7.5999999999999995e-53 < y.im

   1. Initial program 51.9%

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

      1. add-sqr-sqrt 51.9%

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

      2. *-un-lft-identity 51.9%

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

      3. times-frac 51.8%

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

      4. hypot-def 51.8%

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

      5. fma-def 51.8%

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

      6. hypot-def 73.0%

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

   4. Applied egg-rr 73.0%

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

   5. Taylor expanded in y.re around 0 72.8%

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

      1. associate-/l* 72.9%

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

   7. Simplified 72.9%

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

      1. unpow2 72.9%

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

      2. *-un-lft-identity 72.9%

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

      3. times-frac 76.6%

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

   9. Applied egg-rr 76.6%

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

       1. /-rgt-identity 76.6%

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

       2. *-un-lft-identity 76.6%

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

       3. *-commutative 76.6%

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

       4. times-frac 82.0%

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

       5. clear-num 82.0%

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

   11. Applied egg-rr 82.0%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.55 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\frac{y.im \cdot \left(-x.im\right)}{y.re} - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -2.6e+92)
     t_0
     (if (<= y.im -3.2e+32)
       (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im -9e-86)
         (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
         (if (<= y.im 4.3e-75) (/ x.re y.re) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -2.6e+92) {
		tmp = t_0;
	} else if (y_46_im <= -3.2e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -9e-86) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 4.3e-75) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46im <= (-2.6d+92)) then
        tmp = t_0
    else if (y_46im <= (-3.2d+32)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= (-9d-86)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= 4.3d-75) then
        tmp = x_46re / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -2.6e+92) {
		tmp = t_0;
	} else if (y_46_im <= -3.2e+32) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -9e-86) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 4.3e-75) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -2.6e+92:
		tmp = t_0
	elif y_46_im <= -3.2e+32:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= -9e-86:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= 4.3e-75:
		tmp = x_46_re / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.6e+92)
		tmp = t_0;
	elseif (y_46_im <= -3.2e+32)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= -9e-86)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= 4.3e-75)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.6e+92)
		tmp = t_0;
	elseif (y_46_im <= -3.2e+32)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= -9e-86)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= 4.3e-75)
		tmp = x_46_re / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.6e+92], t$95$0, If[LessEqual[y$46$im, -3.2e+32], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9e-86], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.3e-75], N[(x$46$re / y$46$re), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.5999999999999999e92 or 4.2999999999999999e-75 < y.im

   1. Initial program 51.1%

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

      1. add-sqr-sqrt 51.1%

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

      2. *-un-lft-identity 51.1%

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

      3. times-frac 51.0%

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

      4. hypot-def 51.0%

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

      5. fma-def 51.0%

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

      6. hypot-def 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in y.re around 0 73.6%

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

      1. associate-/l* 72.2%

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

   7. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. times-frac 77.5%

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

   9. Applied egg-rr 77.5%

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

       1. /-rgt-identity 77.5%

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

       2. *-un-lft-identity 77.5%

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

       3. *-commutative 77.5%

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

       4. times-frac 82.9%

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

       5. clear-num 82.9%

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

   11. Applied egg-rr 82.9%

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

   if -2.5999999999999999e92 < y.im < -3.1999999999999999e32

   1. Initial program 87.1%

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

   3. Taylor expanded in x.re around 0 87.1%

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

   if -3.1999999999999999e32 < y.im < -8.9999999999999995e-86

   1. Initial program 82.9%

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

      1. add-sqr-sqrt 82.9%

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

      2. *-un-lft-identity 82.9%

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

      3. times-frac 82.9%

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

      4. hypot-def 82.9%

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

      5. fma-def 82.9%

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

      6. hypot-def 86.4%

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

   4. Applied egg-rr 86.4%

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

   5. Taylor expanded in y.re around 0 65.5%

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

      1. associate-/l* 65.4%

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

   7. Simplified 65.4%

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

      1. unpow2 65.4%

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

      2. *-un-lft-identity 65.4%

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

      3. times-frac 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. clear-num 65.4%

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

       2. associate-/r/ 65.4%

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

       3. /-rgt-identity 65.4%

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

       4. *-commutative 65.4%

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

       5. associate-/r* 65.4%

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

       6. clear-num 65.5%

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

   11. Applied egg-rr 65.5%

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

   if -8.9999999999999995e-86 < y.im < 4.2999999999999999e-75

   1. Initial program 80.0%

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

   3. Taylor expanded in y.re around inf 81.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{-85} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.6e-85) (not (<= y.im 1.5e-75)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e-85) || !(y_46_im <= 1.5e-75)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.6d-85)) .or. (.not. (y_46im <= 1.5d-75))) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.6e-85) || !(y_46_im <= 1.5e-75)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.6e-85) or not (y_46_im <= 1.5e-75):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.6e-85) || !(y_46_im <= 1.5e-75))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.6e-85) || ~((y_46_im <= 1.5e-75)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.6e-85], N[Not[LessEqual[y$46$im, 1.5e-75]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.60000000000000033e-85 or 1.4999999999999999e-75 < y.im

   1. Initial program 59.9%

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

      1. add-sqr-sqrt 59.9%

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

      2. *-un-lft-identity 59.9%

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

      3. times-frac 59.8%

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

      4. hypot-def 59.8%

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

      5. fma-def 59.8%

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

      6. hypot-def 76.4%

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

   4. Applied egg-rr 76.4%

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

   5. Taylor expanded in y.re around 0 70.6%

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

      1. associate-/l* 69.6%

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

   7. Simplified 69.6%

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

      1. unpow2 69.6%

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

      2. *-un-lft-identity 69.6%

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

      3. times-frac 73.5%

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

   9. Applied egg-rr 73.5%

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

       1. /-rgt-identity 73.5%

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

       2. *-un-lft-identity 73.5%

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

       3. *-commutative 73.5%

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

       4. times-frac 76.8%

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

       5. clear-num 76.8%

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

   11. Applied egg-rr 76.8%

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

   if -5.60000000000000033e-85 < y.im < 1.4999999999999999e-75

   1. Initial program 80.0%

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

   3. Taylor expanded in y.re around inf 81.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{-85} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-84} \lor \neg \left(y.im \leq 2.9 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.2e-84) (not (<= y.im 2.9e+21)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e-84) || !(y_46_im <= 2.9e+21)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.2d-84)) .or. (.not. (y_46im <= 2.9d+21))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e-84) || !(y_46_im <= 2.9e+21)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.2e-84) or not (y_46_im <= 2.9e+21):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.2e-84) || !(y_46_im <= 2.9e+21))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.2e-84) || ~((y_46_im <= 2.9e+21)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.2e-84], N[Not[LessEqual[y$46$im, 2.9e+21]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.20000000000000009e-84 or 2.9e21 < y.im

   1. Initial program 56.3%

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

   3. Taylor expanded in y.re around 0 64.5%

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

   if -1.20000000000000009e-84 < y.im < 2.9e21

   1. Initial program 82.7%

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

   3. Taylor expanded in y.re around inf 76.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-84} \lor \neg \left(y.im \leq 2.9 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% 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 67.0%

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

3. Taylor expanded in y.re around 0 44.5%

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

4. Final simplification 44.5%

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

Reproduce

herbie shell --seed 2024013 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))