_divideComplex, real part

Percentage Accurate: 61.6% → 83.0%

Time: 10.9s

Alternatives: 11

Speedup: 1.1×

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: 61.6% 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: 83.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im + \frac{x.re \cdot y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e-155)
   (*
    (/ x.im (hypot y.re y.im))
    (/ (+ y.im (/ (* x.re y.re) x.im)) (hypot y.re y.im)))
   (if (<= y.im 7e-114)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 1.8e+29)
       (/
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
        (hypot y.re y.im))
       (/ (+ x.im (* x.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) {
	double tmp;
	if (y_46_im <= -2.1e-155) {
		tmp = (x_46_im / hypot(y_46_re, y_46_im)) * ((y_46_im + ((x_46_re * y_46_re) / x_46_im)) / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 7e-114) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.1e-155)
		tmp = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im + Float64(Float64(x_46_re * y_46_re) / x_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 7e-114)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.1e-155], N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e-114], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+29], N[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.1000000000000002e-155

   1. Initial program 61.3%

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

   3. Taylor expanded in x.im around inf 59.1%

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

      1. associate-/l* 51.4%

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

   5. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. add-sqr-sqrt 51.4%

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

      3. hypot-undefine 51.4%

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

      4. hypot-undefine 51.4%

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

      5. times-frac 86.6%

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

      6. +-commutative 86.6%

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

      7. fma-define 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. fma-undefine 86.6%

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

      2. associate-*r/ 94.3%

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

   9. Applied egg-rr 94.3%

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

   if -2.1000000000000002e-155 < y.im < 7e-114

   1. Initial program 67.3%

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

   3. Taylor expanded in y.re around inf 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-/l* 94.8%

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

      3. fma-define 94.8%

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

   5. Simplified 94.8%

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

   if 7e-114 < y.im < 1.79999999999999988e29

   1. Initial program 87.5%

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

      1. *-un-lft-identity 87.5%

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

      2. add-sqr-sqrt 87.5%

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

      3. times-frac 87.4%

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

      4. hypot-define 87.4%

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

      5. fma-define 87.4%

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

      6. hypot-define 94.0%

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

   4. Applied egg-rr 94.0%

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

      1. associate-*l/ 94.2%

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

      2. *-un-lft-identity 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 1.79999999999999988e29 < y.im

   1. Initial program 49.1%

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

   3. Taylor expanded in y.im around inf 84.8%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im + \frac{x.re \cdot y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{x.im}, y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \end{array} \end{array} \]

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+297)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (/
    1.0
    (/
     (hypot y.re y.im)
     (* x.im (/ (fma x.re (/ y.re x.im) y.im) (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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+297) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (x_46_im * (fma(x_46_re, (y_46_re / x_46_im), y_46_im) / hypot(y_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 (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+297)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(x_46_im * Float64(fma(x_46_re, Float64(y_46_re / x_46_im), y_46_im) / hypot(y_46_re, y_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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+297], N[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$im * N[(N[(x$46$re * N[(y$46$re / x$46$im), $MachinePrecision] + y$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $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))) < 4.9999999999999998e297

   1. Initial program 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. times-frac 82.0%

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

      4. hypot-define 82.0%

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

      5. fma-define 82.0%

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

      6. hypot-define 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 4.9999999999999998e297 < (/.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 15.4%

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

   3. Taylor expanded in x.im around inf 15.4%

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

      1. associate-/l* 15.4%

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

   5. Simplified 15.4%

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

      1. *-commutative 15.4%

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

      2. add-sqr-sqrt 15.4%

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

      3. hypot-undefine 15.4%

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

      4. hypot-undefine 15.4%

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

      5. times-frac 71.5%

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

      6. +-commutative 71.5%

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

      7. fma-define 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 65.9%

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

   9. Applied egg-rr 65.9%

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

       1. associate-*r/ 67.7%

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

       2. clear-num 67.5%

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

       3. associate-/l* 73.2%

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

       4. fma-define 73.2%

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

   11. Applied egg-rr 73.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{x.im}, y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{x.im}, y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+297)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (*
    (/ (fma x.re (/ y.re x.im) y.im) (hypot y.re y.im))
    (/ x.im (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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+297) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (fma(x_46_re, (y_46_re / x_46_im), y_46_im) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_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 (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+297)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(fma(x_46_re, Float64(y_46_re / x_46_im), y_46_im) / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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+297], N[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * N[(y$46$re / x$46$im), $MachinePrecision] + y$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $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))) < 4.9999999999999998e297

   1. Initial program 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. add-sqr-sqrt 82.0%

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

      3. times-frac 82.0%

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

      4. hypot-define 82.0%

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

      5. fma-define 82.0%

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

      6. hypot-define 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-un-lft-identity 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 4.9999999999999998e297 < (/.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 15.4%

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

   3. Taylor expanded in x.im around inf 15.4%

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

      1. associate-/l* 15.4%

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

   5. Simplified 15.4%

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

      1. *-commutative 15.4%

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

      2. add-sqr-sqrt 15.4%

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

      3. hypot-undefine 15.4%

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

      4. hypot-undefine 15.4%

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

      5. times-frac 71.5%

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

      6. +-commutative 71.5%

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

      7. fma-define 71.5%

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

   7. Applied egg-rr 71.5%

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

Alternative 4: 82.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im + \frac{x.re \cdot y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.12e-159)
   (*
    (/ x.im (hypot y.re y.im))
    (/ (+ y.im (/ (* x.re y.re) x.im)) (hypot y.re y.im)))
   (if (<= y.im 6.2e-119)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 1.65e+29)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (+ x.im (* x.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) {
	double tmp;
	if (y_46_im <= -1.12e-159) {
		tmp = (x_46_im / hypot(y_46_re, y_46_im)) * ((y_46_im + ((x_46_re * y_46_re) / x_46_im)) / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 6.2e-119) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.65e+29) {
		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));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.12e-159)
		tmp = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im + Float64(Float64(x_46_re * y_46_re) / x_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 6.2e-119)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.65e+29)
		tmp = 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)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.12e-159], N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.2e-119], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+29], 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], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.12000000000000006e-159

   1. Initial program 61.3%

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

   3. Taylor expanded in x.im around inf 59.1%

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

      1. associate-/l* 51.4%

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

   5. Simplified 51.4%

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

      1. *-commutative 51.4%

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

      2. add-sqr-sqrt 51.4%

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

      3. hypot-undefine 51.4%

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

      4. hypot-undefine 51.4%

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

      5. times-frac 86.6%

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

      6. +-commutative 86.6%

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

      7. fma-define 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. fma-undefine 86.6%

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

      2. associate-*r/ 94.3%

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

   9. Applied egg-rr 94.3%

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

   if -1.12000000000000006e-159 < y.im < 6.19999999999999956e-119

   1. Initial program 67.3%

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

   3. Taylor expanded in y.re around inf 94.6%

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

      1. +-commutative 94.6%

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

      2. associate-/l* 94.8%

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

      3. fma-define 94.8%

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

   5. Simplified 94.8%

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

   if 6.19999999999999956e-119 < y.im < 1.64999999999999992e29

   1. Initial program 87.5%

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

   if 1.64999999999999992e29 < y.im

   1. Initial program 49.1%

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

   3. Taylor expanded in y.im around inf 84.8%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im + \frac{x.re \cdot y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

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)))))
   (if (<= y.im -4.8e+47)
     (* (/ x.im (hypot y.re y.im)) (/ y.im (hypot y.re y.im)))
     (if (<= y.im -2e-152)
       t_0
       (if (<= y.im 3.1e-118)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.8e+29)
           t_0
           (/ (+ x.im (* x.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) {
	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 tmp;
	if (y_46_im <= -4.8e+47) {
		tmp = (x_46_im / hypot(y_46_re, y_46_im)) * (y_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -2e-152) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-118) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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)))
	tmp = 0.0
	if (y_46_im <= -4.8e+47)
		tmp = Float64(Float64(x_46_im / hypot(y_46_re, y_46_im)) * Float64(y_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -2e-152)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-118)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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]}, If[LessEqual[y$46$im, -4.8e+47], N[(N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2e-152], t$95$0, If[LessEqual[y$46$im, 3.1e-118], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+29], t$95$0, N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.80000000000000037e47

   1. Initial program 43.3%

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

   3. Taylor expanded in x.im around inf 41.6%

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

      1. associate-/l* 32.8%

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

   5. Simplified 32.8%

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

      1. *-commutative 32.8%

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

      2. add-sqr-sqrt 32.8%

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

      3. hypot-undefine 32.8%

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

      4. hypot-undefine 32.8%

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

      5. times-frac 84.1%

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

      6. +-commutative 84.1%

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

      7. fma-define 84.1%

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

   7. Applied egg-rr 84.1%

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

   8. Taylor expanded in x.re around 0 89.1%

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

   if -4.80000000000000037e47 < y.im < -2.00000000000000013e-152 or 3.1000000000000001e-118 < y.im < 1.79999999999999988e29

   1. Initial program 89.1%

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

   if -2.00000000000000013e-152 < y.im < 3.1000000000000001e-118

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. +-commutative 94.7%

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

      2. associate-/l* 94.9%

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

      3. fma-define 94.9%

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

   5. Simplified 94.9%

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

   if 1.79999999999999988e29 < y.im

   1. Initial program 49.1%

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

   3. Taylor expanded in y.im around inf 84.8%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -5.9 \cdot 10^{-149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -3.8e+138)
     t_1
     (if (<= y.im -5.9e-149)
       t_0
       (if (<= y.im 1.55e-116)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 1.8e+29) 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 + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -3.8e+138) {
		tmp = t_1;
	} else if (y_46_im <= -5.9e-149) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e-116) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 1.8e+29) {
		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 + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.8e+138)
		tmp = t_1;
	elseif (y_46_im <= -5.9e-149)
		tmp = t_0;
	elseif (y_46_im <= 1.55e-116)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 1.8e+29)
		tmp = t_0;
	else
		tmp = 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[(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 + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+138], t$95$1, If[LessEqual[y$46$im, -5.9e-149], t$95$0, If[LessEqual[y$46$im, 1.55e-116], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.80000000000000012e138 or 1.79999999999999988e29 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 86.3%

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

      1. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

   if -3.80000000000000012e138 < y.im < -5.9000000000000002e-149 or 1.55000000000000009e-116 < y.im < 1.79999999999999988e29

   1. Initial program 87.5%

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

   if -5.9000000000000002e-149 < y.im < 1.55000000000000009e-116

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. +-commutative 94.7%

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

      2. associate-/l* 94.9%

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

      3. fma-define 94.9%

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

   5. Simplified 94.9%

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

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.95 \cdot 10^{-151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -1.45e+128)
     t_1
     (if (<= y.im -2.95e-151)
       t_0
       (if (<= y.im 1.4e-116)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 1.65e+29) 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 + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.45e+128) {
		tmp = t_1;
	} else if (y_46_im <= -2.95e-151) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-116) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.65e+29) {
		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 + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-1.45d+128)) then
        tmp = t_1
    else if (y_46im <= (-2.95d-151)) then
        tmp = t_0
    else if (y_46im <= 1.4d-116) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 1.65d+29) 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 + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.45e+128) {
		tmp = t_1;
	} else if (y_46_im <= -2.95e-151) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-116) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.65e+29) {
		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 + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -1.45e+128:
		tmp = t_1
	elif y_46_im <= -2.95e-151:
		tmp = t_0
	elif y_46_im <= 1.4e-116:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 1.65e+29:
		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 + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.45e+128)
		tmp = t_1;
	elseif (y_46_im <= -2.95e-151)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-116)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.65e+29)
		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 + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.45e+128)
		tmp = t_1;
	elseif (y_46_im <= -2.95e-151)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-116)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 1.65e+29)
		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 + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.45e+128], t$95$1, If[LessEqual[y$46$im, -2.95e-151], t$95$0, If[LessEqual[y$46$im, 1.4e-116], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.45e128 or 1.64999999999999992e29 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 86.3%

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

      1. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

   if -1.45e128 < y.im < -2.95e-151 or 1.3999999999999999e-116 < y.im < 1.64999999999999992e29

   1. Initial program 87.5%

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

   if -2.95e-151 < y.im < 1.3999999999999999e-116

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around inf 94.7%

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

      1. associate-/l* 94.9%

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

   5. Simplified 94.9%

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

Alternative 8: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{-10} \lor \neg \left(y.im \leq 3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.4e-10) (not (<= y.im 3e-7)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.4e-10) || !(y_46_im <= 3e-7)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.4d-10)) .or. (.not. (y_46im <= 3d-7))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.4e-10) || !(y_46_im <= 3e-7)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.4e-10) or not (y_46_im <= 3e-7):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.4e-10) || !(y_46_im <= 3e-7))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.4e-10) || ~((y_46_im <= 3e-7)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.4e-10], N[Not[LessEqual[y$46$im, 3e-7]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.3999999999999998e-10 or 2.9999999999999999e-7 < y.im

   1. Initial program 53.3%

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

   3. Taylor expanded in y.im around inf 82.9%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   if -4.3999999999999998e-10 < y.im < 2.9999999999999999e-7

   1. Initial program 74.8%

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

   3. Taylor expanded in y.re around inf 87.4%

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

      1. associate-/l* 87.5%

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

   5. Simplified 87.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{-10} \lor \neg \left(y.im \leq 3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 550000000000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7e+36)
   (/ 1.0 (/ y.re x.re))
   (if (<= y.re 550000000000.0)
     (/ (+ x.im (* x.re (/ y.re y.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_re <= -7e+36) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 550000000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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_46re <= (-7d+36)) then
        tmp = 1.0d0 / (y_46re / x_46re)
    else if (y_46re <= 550000000000.0d0) then
        tmp = (x_46im + (x_46re * (y_46re / y_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_re <= -7e+36) {
		tmp = 1.0 / (y_46_re / x_46_re);
	} else if (y_46_re <= 550000000000.0) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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_re <= -7e+36:
		tmp = 1.0 / (y_46_re / x_46_re)
	elif y_46_re <= 550000000000.0:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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_re <= -7e+36)
		tmp = Float64(1.0 / Float64(y_46_re / x_46_re));
	elseif (y_46_re <= 550000000000.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_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_re <= -7e+36)
		tmp = 1.0 / (y_46_re / x_46_re);
	elseif (y_46_re <= 550000000000.0)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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[LessEqual[y$46$re, -7e+36], N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 550000000000.0], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.9999999999999996e36

   1. Initial program 50.1%

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

   3. Taylor expanded in y.re around inf 63.7%

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

      1. clear-num 65.0%

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

      2. inv-pow 65.0%

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

   5. Applied egg-rr 65.0%

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

      1. unpow-1 65.0%

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

   7. Simplified 65.0%

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

   if -6.9999999999999996e36 < y.re < 5.5e11

   1. Initial program 68.9%

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

   3. Taylor expanded in y.im around inf 84.6%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

   if 5.5e11 < y.re

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 72.3%

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

Alternative 10: 63.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{-52} \lor \neg \left(y.im \leq 7.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6e-52) (not (<= y.im 7.5e-8))) (/ 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 <= -6e-52) || !(y_46_im <= 7.5e-8)) {
		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 <= (-6d-52)) .or. (.not. (y_46im <= 7.5d-8))) 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 <= -6e-52) || !(y_46_im <= 7.5e-8)) {
		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 <= -6e-52) or not (y_46_im <= 7.5e-8):
		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 <= -6e-52) || !(y_46_im <= 7.5e-8))
		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 <= -6e-52) || ~((y_46_im <= 7.5e-8)))
		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, -6e-52], N[Not[LessEqual[y$46$im, 7.5e-8]], $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 < -6e-52 or 7.4999999999999997e-8 < y.im

   1. Initial program 55.0%

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

   3. Taylor expanded in y.re around 0 73.1%

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

   if -6e-52 < y.im < 7.4999999999999997e-8

   1. Initial program 74.2%

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

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

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{-52} \lor \neg \left(y.im \leq 7.5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.5% 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 64.1%

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

3. Taylor expanded in y.re around 0 47.6%

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

Reproduce

herbie shell --seed 2024132 
(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))))