_divideComplex, real part

Percentage Accurate: 62.0% → 88.7%

Time: 13.4s

Alternatives: 14

Speedup: 0.8×

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: 62.0% 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: 88.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im + x.re \cdot \frac{y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

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)))
      2e+251)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (*
    (/ (+ y.im (* x.re (/ y.re x.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))) <= 2e+251) {
		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 = ((y_46_im + (x_46_re * (y_46_re / x_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))) <= 2e+251)
		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(Float64(y_46_im + Float64(x_46_re * Float64(y_46_re / x_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], 2e+251], 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[(N[(y$46$im + N[(x$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $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))) < 2.0000000000000001e251

   1. Initial program 76.2%

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

      1. fma-define 76.2%

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

      2. fma-define 76.2%

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

   3. Simplified 76.2%

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

      1. *-un-lft-identity 76.2%

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

      2. fma-define 76.2%

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

      3. add-sqr-sqrt 76.2%

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

      4. times-frac 76.3%

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

      5. fma-define 76.3%

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

      6. hypot-define 76.3%

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

      7. fma-define 76.3%

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

      8. fma-define 76.3%

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

      9. hypot-define 96.1%

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

   6. Applied egg-rr 96.1%

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

   if 2.0000000000000001e251 < (/.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 14.9%

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

      1. fma-define 14.9%

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

      2. fma-define 14.9%

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

   3. Simplified 14.9%

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

   5. Taylor expanded in x.im around inf 14.9%

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

      1. associate-/l* 13.6%

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

   7. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. fma-undefine 13.6%

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

      3. add-sqr-sqrt 13.6%

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

      4. hypot-undefine 13.6%

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

      5. hypot-undefine 13.6%

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

      6. times-frac 75.6%

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

      7. +-commutative 75.6%

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

      8. fma-define 75.6%

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

   9. Applied egg-rr 75.6%

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

       1. fma-undefine 75.6%

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

   11. Applied egg-rr 75.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im + x.re \cdot \frac{y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.4e+75)
   (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
   (if (<= y.re -6.8e-114)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 1.26e-50)
       (/ (+ x.im (* y.re (/ x.re y.im))) y.im)
       (if (<= y.re 3e+142)
         (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
         (/ (+ x.re (* x.im (/ y.im y.re))) (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.4e+75) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -6.8e-114) {
		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 if (y_46_re <= 1.26e-50) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 3e+142) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_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))) / 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 (y_46_re <= -1.4e+75)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -6.8e-114)
		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)));
	elseif (y_46_re <= 1.26e-50)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	elseif (y_46_re <= 3e+142)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / 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[y$46$re, -1.4e+75], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.8e-114], 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$46$re, 1.26e-50], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3e+142], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.40000000000000006e75

   1. Initial program 30.5%

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

      1. fma-define 30.5%

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

      2. fma-define 30.5%

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

   3. Simplified 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. fma-define 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.5%

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

      5. fma-define 30.5%

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

      6. hypot-define 30.5%

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

      7. fma-define 30.5%

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

      8. fma-define 30.5%

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

      9. hypot-define 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in y.re around -inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in y.re around -inf 80.4%

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

       1. clear-num 80.5%

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

       2. un-div-inv 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. associate-/r/ 83.7%

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

   14. Simplified 83.7%

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

   if -1.40000000000000006e75 < y.re < -6.79999999999999962e-114

   1. Initial program 89.7%

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

   if -6.79999999999999962e-114 < y.re < 1.26e-50

   1. Initial program 64.4%

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

      1. fma-define 64.4%

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

      2. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in y.im around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. times-frac 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 1.26e-50 < y.re < 2.99999999999999975e142

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.5%

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

   3. Simplified 69.5%

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

   if 2.99999999999999975e142 < y.re

   1. Initial program 23.1%

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

      1. fma-define 23.1%

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

      2. fma-define 23.1%

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

   3. Simplified 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. fma-define 23.1%

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

      3. add-sqr-sqrt 23.1%

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

      4. times-frac 23.1%

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

      5. fma-define 23.1%

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

      6. hypot-define 23.1%

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

      7. fma-define 23.1%

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

      8. fma-define 23.1%

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

      9. hypot-define 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in y.re around -inf 23.2%

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

      1. distribute-lft-out 23.2%

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

      2. associate-/l* 23.2%

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

   9. Simplified 23.2%

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

       1. associate-*l/ 23.2%

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

       2. *-un-lft-identity 23.2%

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

       3. add-sqr-sqrt 10.6%

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

       4. sqrt-unprod 54.8%

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

       5. mul-1-neg 54.8%

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

       6. mul-1-neg 54.8%

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

       7. sqr-neg 54.8%

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

       8. sqrt-unprod 59.0%

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

       9. add-sqr-sqrt 97.6%

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

       10. +-commutative 97.6%

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

       11. fma-define 97.6%

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

   11. Applied egg-rr 97.6%

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

       1. fma-undefine 97.6%

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

   13. Applied egg-rr 97.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im + x.re \cdot \frac{y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.5e+75)
   (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
   (if (<= y.re -3.8e-106)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 7.8e+115)
       (*
        (/ (+ y.im (* x.re (/ y.re x.im))) (hypot y.re y.im))
        (/ x.im (hypot y.re y.im)))
       (/ (+ x.re (* x.im (/ y.im y.re))) (hypot y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.5e+75) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -3.8e-106) {
		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 if (y_46_re <= 7.8e+115) {
		tmp = ((y_46_im + (x_46_re * (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.5e+75) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -3.8e-106) {
		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 if (y_46_re <= 7.8e+115) {
		tmp = ((y_46_im + (x_46_re * (y_46_re / x_46_im))) / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5.5e+75:
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= -3.8e-106:
		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))
	elif y_46_re <= 7.8e+115:
		tmp = ((y_46_im + (x_46_re * (y_46_re / x_46_im))) / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.5e+75)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -3.8e-106)
		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)));
	elseif (y_46_re <= 7.8e+115)
		tmp = Float64(Float64(Float64(y_46_im + Float64(x_46_re * Float64(y_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5.5e+75)
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= -3.8e-106)
		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));
	elseif (y_46_re <= 7.8e+115)
		tmp = ((y_46_im + (x_46_re * (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.5e+75], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.8e-106], 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$46$re, 7.8e+115], N[(N[(N[(y$46$im + N[(x$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $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], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.5000000000000001e75

   1. Initial program 30.5%

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

      1. fma-define 30.5%

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

      2. fma-define 30.5%

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

   3. Simplified 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. fma-define 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.5%

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

      5. fma-define 30.5%

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

      6. hypot-define 30.5%

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

      7. fma-define 30.5%

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

      8. fma-define 30.5%

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

      9. hypot-define 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in y.re around -inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in y.re around -inf 80.4%

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

       1. clear-num 80.5%

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

       2. un-div-inv 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. associate-/r/ 83.7%

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

   14. Simplified 83.7%

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

   if -5.5000000000000001e75 < y.re < -3.7999999999999999e-106

   1. Initial program 90.4%

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

   if -3.7999999999999999e-106 < y.re < 7.80000000000000012e115

   1. Initial program 65.6%

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

      1. fma-define 65.6%

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

      2. fma-define 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in x.im around inf 61.2%

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

      1. associate-/l* 59.8%

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

   7. Simplified 59.8%

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

      1. *-commutative 59.8%

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

      2. fma-undefine 59.8%

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

      3. add-sqr-sqrt 59.8%

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

      4. hypot-undefine 59.8%

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

      5. hypot-undefine 59.8%

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

      6. times-frac 91.2%

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

      7. +-commutative 91.2%

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

      8. fma-define 91.2%

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

   9. Applied egg-rr 91.2%

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

       1. fma-undefine 91.2%

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

   11. Applied egg-rr 91.2%

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

   if 7.80000000000000012e115 < y.re

   1. Initial program 29.0%

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

      1. fma-define 29.0%

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

      2. fma-define 29.0%

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

   3. Simplified 29.0%

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

      1. *-un-lft-identity 29.0%

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

      2. fma-define 29.0%

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

      3. add-sqr-sqrt 29.0%

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

      4. times-frac 29.0%

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

      5. fma-define 29.0%

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

      6. hypot-define 29.0%

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

      7. fma-define 29.0%

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

      8. fma-define 29.0%

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

      9. hypot-define 67.3%

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

   6. Applied egg-rr 67.3%

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

   7. Taylor expanded in y.re around -inf 23.7%

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

      1. distribute-lft-out 23.7%

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

      2. associate-/l* 23.8%

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

   9. Simplified 23.8%

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

       1. associate-*l/ 23.8%

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

       2. *-un-lft-identity 23.8%

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

       3. add-sqr-sqrt 12.4%

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

       4. sqrt-unprod 54.6%

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

       5. mul-1-neg 54.6%

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

       6. mul-1-neg 54.6%

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

       7. sqr-neg 54.6%

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

       8. sqrt-unprod 55.4%

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

       9. add-sqr-sqrt 92.6%

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

       10. +-commutative 92.6%

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

       11. fma-define 92.6%

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

   11. Applied egg-rr 92.6%

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

       1. fma-undefine 92.6%

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

   13. Applied egg-rr 92.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im + x.re \cdot \frac{y.re}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.1 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -5.5e+72)
     (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
     (if (<= y.re -6.1e-114)
       t_0
       (if (<= y.re 3.25e-52)
         (/ (+ x.im (* y.re (/ x.re y.im))) y.im)
         (if (<= y.re 3.5e+142)
           t_0
           (/ (+ x.re (* x.im (/ y.im y.re))) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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_re <= -5.5e+72) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -6.1e-114) {
		tmp = t_0;
	} else if (y_46_re <= 3.25e-52) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 3.5e+142) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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_re <= -5.5e+72) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -6.1e-114) {
		tmp = t_0;
	} else if (y_46_re <= 3.25e-52) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 3.5e+142) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -5.5e+72:
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= -6.1e-114:
		tmp = t_0
	elif y_46_re <= 3.25e-52:
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im
	elif y_46_re <= 3.5e+142:
		tmp = t_0
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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_re <= -5.5e+72)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -6.1e-114)
		tmp = t_0;
	elseif (y_46_re <= 3.25e-52)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	elseif (y_46_re <= 3.5e+142)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -5.5e+72)
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= -6.1e-114)
		tmp = t_0;
	elseif (y_46_re <= 3.25e-52)
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	elseif (y_46_re <= 3.5e+142)
		tmp = t_0;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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$46$re, -5.5e+72], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.1e-114], t$95$0, If[LessEqual[y$46$re, 3.25e-52], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+142], t$95$0, N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.5e72

   1. Initial program 30.5%

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

      1. fma-define 30.5%

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

      2. fma-define 30.5%

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

   3. Simplified 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. fma-define 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.5%

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

      5. fma-define 30.5%

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

      6. hypot-define 30.5%

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

      7. fma-define 30.5%

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

      8. fma-define 30.5%

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

      9. hypot-define 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in y.re around -inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in y.re around -inf 80.4%

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

       1. clear-num 80.5%

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

       2. un-div-inv 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. associate-/r/ 83.7%

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

   14. Simplified 83.7%

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

   if -5.5e72 < y.re < -6.09999999999999977e-114 or 3.25e-52 < y.re < 3.49999999999999997e142

   1. Initial program 80.2%

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

   if -6.09999999999999977e-114 < y.re < 3.25e-52

   1. Initial program 64.4%

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

      1. fma-define 64.4%

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

      2. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in y.im around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. times-frac 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 3.49999999999999997e142 < y.re

   1. Initial program 23.1%

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

      1. fma-define 23.1%

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

      2. fma-define 23.1%

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

   3. Simplified 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. fma-define 23.1%

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

      3. add-sqr-sqrt 23.1%

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

      4. times-frac 23.1%

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

      5. fma-define 23.1%

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

      6. hypot-define 23.1%

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

      7. fma-define 23.1%

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

      8. fma-define 23.1%

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

      9. hypot-define 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in y.re around -inf 23.2%

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

      1. distribute-lft-out 23.2%

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

      2. associate-/l* 23.2%

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

   9. Simplified 23.2%

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

       1. associate-*l/ 23.2%

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

       2. *-un-lft-identity 23.2%

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

       3. add-sqr-sqrt 10.6%

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

       4. sqrt-unprod 54.8%

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

       5. mul-1-neg 54.8%

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

       6. mul-1-neg 54.8%

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

       7. sqr-neg 54.8%

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

       8. sqrt-unprod 59.0%

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

       9. add-sqr-sqrt 97.6%

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

       10. +-commutative 97.6%

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

       11. fma-define 97.6%

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

   11. Applied egg-rr 97.6%

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

       1. fma-undefine 97.6%

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

   13. Applied egg-rr 97.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+142}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.65e+76)
     (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
     (if (<= y.re -6.8e-114)
       t_0
       (if (<= y.re 3.25e-52)
         (/ (+ x.im (* y.re (/ x.re y.im))) y.im)
         (if (<= y.re 8e+142) t_0 (/ (fma x.im (/ y.im y.re) x.re) 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)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.65e+76) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -6.8e-114) {
		tmp = t_0;
	} else if (y_46_re <= 3.25e-52) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 8e+142) {
		tmp = t_0;
	} else {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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_re <= -2.65e+76)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -6.8e-114)
		tmp = t_0;
	elseif (y_46_re <= 3.25e-52)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	elseif (y_46_re <= 8e+142)
		tmp = t_0;
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	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$46$re, -2.65e+76], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.8e-114], t$95$0, If[LessEqual[y$46$re, 3.25e-52], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8e+142], t$95$0, N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.65000000000000008e76

   1. Initial program 30.5%

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

      1. fma-define 30.5%

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

      2. fma-define 30.5%

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

   3. Simplified 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. fma-define 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.5%

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

      5. fma-define 30.5%

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

      6. hypot-define 30.5%

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

      7. fma-define 30.5%

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

      8. fma-define 30.5%

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

      9. hypot-define 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in y.re around -inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in y.re around -inf 80.4%

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

       1. clear-num 80.5%

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

       2. un-div-inv 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. associate-/r/ 83.7%

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

   14. Simplified 83.7%

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

   if -2.65000000000000008e76 < y.re < -6.79999999999999962e-114 or 3.25e-52 < y.re < 8.00000000000000041e142

   1. Initial program 80.2%

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

   if -6.79999999999999962e-114 < y.re < 3.25e-52

   1. Initial program 64.4%

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

      1. fma-define 64.4%

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

      2. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in y.im around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. times-frac 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 8.00000000000000041e142 < y.re

   1. Initial program 23.1%

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

      1. fma-define 23.1%

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

      2. fma-define 23.1%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in y.re around inf 85.3%

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

      1. +-commutative 85.3%

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

      2. associate-/l* 97.5%

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

      3. fma-define 97.5%

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

   7. Simplified 97.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.25 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.35e+73)
     (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
     (if (<= y.re -5.2e-114)
       t_0
       (if (<= y.re 6.6e-52)
         (/ (+ x.im (* y.re (/ x.re y.im))) y.im)
         (if (<= y.re 2.4e+144)
           t_0
           (/ (+ 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 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_re <= -1.35e+73) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -5.2e-114) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-52) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 2.4e+144) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1.35d+73)) then
        tmp = ((-1.0d0) / y_46re) * (-x_46re - (y_46im * (x_46im / y_46re)))
    else if (y_46re <= (-5.2d-114)) then
        tmp = t_0
    else if (y_46re <= 6.6d-52) then
        tmp = (x_46im + (y_46re * (x_46re / y_46im))) / y_46im
    else if (y_46re <= 2.4d+144) then
        tmp = t_0
    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 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_re <= -1.35e+73) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -5.2e-114) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-52) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	} else if (y_46_re <= 2.4e+144) {
		tmp = t_0;
	} 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):
	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))
	tmp = 0
	if y_46_re <= -1.35e+73:
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= -5.2e-114:
		tmp = t_0
	elif y_46_re <= 6.6e-52:
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im
	elif y_46_re <= 2.4e+144:
		tmp = t_0
	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)
	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_re <= -1.35e+73)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -5.2e-114)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-52)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	elseif (y_46_re <= 2.4e+144)
		tmp = t_0;
	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)
	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));
	tmp = 0.0;
	if (y_46_re <= -1.35e+73)
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= -5.2e-114)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-52)
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	elseif (y_46_re <= 2.4e+144)
		tmp = t_0;
	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_] := 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$46$re, -1.35e+73], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-114], t$95$0, If[LessEqual[y$46$re, 6.6e-52], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+144], t$95$0, 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 4 regimes

2. if y.re < -1.35e73

   1. Initial program 30.5%

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

      1. fma-define 30.5%

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

      2. fma-define 30.5%

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

   3. Simplified 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. fma-define 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.5%

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

      5. fma-define 30.5%

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

      6. hypot-define 30.5%

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

      7. fma-define 30.5%

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

      8. fma-define 30.5%

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

      9. hypot-define 52.3%

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

   6. Applied egg-rr 52.3%

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

   7. Taylor expanded in y.re around -inf 75.1%

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

      1. distribute-lft-out 75.1%

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

      2. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in y.re around -inf 80.4%

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

       1. clear-num 80.5%

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

       2. un-div-inv 80.4%

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

   12. Applied egg-rr 80.4%

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

       1. associate-/r/ 83.7%

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

   14. Simplified 83.7%

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

   if -1.35e73 < y.re < -5.20000000000000026e-114 or 6.5999999999999999e-52 < y.re < 2.4000000000000001e144

   1. Initial program 80.2%

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

   if -5.20000000000000026e-114 < y.re < 6.5999999999999999e-52

   1. Initial program 64.4%

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

      1. fma-define 64.4%

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

      2. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in y.im around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. *-un-lft-identity 89.7%

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

      3. times-frac 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 2.4000000000000001e144 < y.re

   1. Initial program 23.1%

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

      1. fma-define 23.1%

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

      2. fma-define 23.1%

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

   3. Simplified 23.1%

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

      1. *-un-lft-identity 23.1%

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

      2. fma-define 23.1%

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

      3. add-sqr-sqrt 23.1%

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

      4. times-frac 23.1%

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

      5. fma-define 23.1%

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

      6. hypot-define 23.1%

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

      7. fma-define 23.1%

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

      8. fma-define 23.1%

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

      9. hypot-define 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in y.re around inf 85.3%

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

      1. associate-/l* 97.5%

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

   9. Simplified 97.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1950000 \lor \neg \left(y.re \leq 2.6 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1950000.0) (not (<= y.re 2.6e-50)))
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
   (/ (+ 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_re <= -1950000.0) || !(y_46_re <= 2.6e-50)) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1950000.0d0)) .or. (.not. (y_46re <= 2.6d-50))) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1950000.0) || !(y_46_re <= 2.6e-50)) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1950000.0) or not (y_46_re <= 2.6e-50):
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	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_re <= -1950000.0) || !(y_46_re <= 2.6e-50))
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1950000.0) || ~((y_46_re <= 2.6e-50)))
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1950000.0], N[Not[LessEqual[y$46$re, 2.6e-50]], $MachinePrecision]], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.95e6 or 2.6000000000000001e-50 < y.re

   1. Initial program 46.5%

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

      1. fma-define 46.5%

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

      2. fma-define 46.5%

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

   3. Simplified 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. fma-define 46.5%

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

      3. add-sqr-sqrt 46.5%

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

      4. times-frac 46.5%

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

      5. fma-define 46.5%

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

      6. hypot-define 46.5%

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

      7. fma-define 46.5%

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

      8. fma-define 46.5%

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

      9. hypot-define 68.5%

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

   6. Applied egg-rr 68.5%

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

   7. Taylor expanded in y.re around inf 70.3%

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

      1. associate-/l* 76.0%

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

   9. Simplified 76.0%

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

   if -1.95e6 < y.re < 2.6000000000000001e-50

   1. Initial program 69.6%

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

      1. fma-define 69.6%

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

      2. fma-define 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in y.im around inf 85.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1950000 \lor \neg \left(y.re \leq 2.6 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2400000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2400000.0) (not (<= y.re 3.5e+116)))
   (/ x.re y.re)
   (/ (+ 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_re <= -2400000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2400000.0d0)) .or. (.not. (y_46re <= 3.5d+116))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2400000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2400000.0) or not (y_46_re <= 3.5e+116):
		tmp = x_46_re / y_46_re
	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_re <= -2400000.0) || !(y_46_re <= 3.5e+116))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2400000.0) || ~((y_46_re <= 3.5e+116)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2400000.0], N[Not[LessEqual[y$46$re, 3.5e+116]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.4e6 or 3.49999999999999997e116 < y.re

   1. Initial program 39.7%

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

      1. fma-define 39.7%

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

      2. fma-define 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in y.re around inf 70.1%

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

   if -2.4e6 < y.re < 3.49999999999999997e116

   1. Initial program 69.0%

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

      1. fma-define 69.0%

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

      2. fma-define 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in y.im around inf 78.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2400000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2200000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2200000.0) (not (<= y.re 3.5e+116)))
   (/ x.re y.re)
   (/ (+ x.im (/ x.re (/ y.im 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 ((y_46_re <= -2200000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2200000.0d0)) .or. (.not. (y_46re <= 3.5d+116))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2200000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2200000.0) or not (y_46_re <= 3.5e+116):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / 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 ((y_46_re <= -2200000.0) || !(y_46_re <= 3.5e+116))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2200000.0) || ~((y_46_re <= 3.5e+116)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2200000.0], N[Not[LessEqual[y$46$re, 3.5e+116]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.2e6 or 3.49999999999999997e116 < y.re

   1. Initial program 39.7%

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

      1. fma-define 39.7%

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

      2. fma-define 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in y.re around inf 70.1%

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

   if -2.2e6 < y.re < 3.49999999999999997e116

   1. Initial program 69.0%

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

      1. fma-define 69.0%

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

      2. fma-define 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in y.im around inf 78.0%

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

      1. associate-/l* 77.5%

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

   7. Simplified 77.5%

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

      1. clear-num 77.5%

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

      2. un-div-inv 77.5%

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

   9. Applied egg-rr 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2200000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2200000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2200000.0) (not (<= y.re 3.5e+116)))
   (/ x.re y.re)
   (/ (+ 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_re <= -2200000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2200000.0d0)) .or. (.not. (y_46re <= 3.5d+116))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2200000.0) || !(y_46_re <= 3.5e+116)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2200000.0) or not (y_46_re <= 3.5e+116):
		tmp = x_46_re / y_46_re
	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_re <= -2200000.0) || !(y_46_re <= 3.5e+116))
		tmp = Float64(x_46_re / y_46_re);
	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

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2200000.0) || ~((y_46_re <= 3.5e+116)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2200000.0], N[Not[LessEqual[y$46$re, 3.5e+116]], $MachinePrecision]], N[(x$46$re / y$46$re), $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 2 regimes

2. if y.re < -2.2e6 or 3.49999999999999997e116 < y.re

   1. Initial program 39.7%

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

      1. fma-define 39.7%

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

      2. fma-define 39.7%

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

   3. Simplified 39.7%

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

   5. Taylor expanded in y.re around inf 70.1%

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

   if -2.2e6 < y.re < 3.49999999999999997e116

   1. Initial program 69.0%

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

      1. fma-define 69.0%

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

      2. fma-define 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in y.im around inf 78.0%

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

      1. associate-/l* 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2200000 \lor \neg \left(y.re \leq 3.5 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2400000:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2400000.0)
   (* (/ -1.0 y.re) (- (- x.re) (* y.im (/ x.im y.re))))
   (if (<= y.re 3.75e-47)
     (/ (+ x.im (* y.re (/ x.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_re <= -2400000.0) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= 3.75e-47) {
		tmp = (x_46_im + (y_46_re * (x_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_46re <= (-2400000.0d0)) then
        tmp = ((-1.0d0) / y_46re) * (-x_46re - (y_46im * (x_46im / y_46re)))
    else if (y_46re <= 3.75d-47) then
        tmp = (x_46im + (y_46re * (x_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_re <= -2400000.0) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= 3.75e-47) {
		tmp = (x_46_im + (y_46_re * (x_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_re <= -2400000.0:
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= 3.75e-47:
		tmp = (x_46_im + (y_46_re * (x_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_re <= -2400000.0)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= 3.75e-47)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_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_re <= -2400000.0)
		tmp = (-1.0 / y_46_re) * (-x_46_re - (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= 3.75e-47)
		tmp = (x_46_im + (y_46_re * (x_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[LessEqual[y$46$re, -2400000.0], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.75e-47], N[(N[(x$46$im + N[(y$46$re * N[(x$46$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 3 regimes

2. if y.re < -2.4e6

   1. Initial program 45.3%

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

      1. fma-define 45.3%

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

      2. fma-define 45.3%

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

   3. Simplified 45.3%

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

      1. *-un-lft-identity 45.3%

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

      2. fma-define 45.3%

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

      3. add-sqr-sqrt 45.3%

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

      4. times-frac 45.4%

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

      5. fma-define 45.4%

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

      6. hypot-define 45.4%

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

      7. fma-define 45.4%

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

      8. fma-define 45.4%

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

      9. hypot-define 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in y.re around -inf 73.8%

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

      1. distribute-lft-out 73.8%

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

      2. associate-/l* 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in y.re around -inf 77.7%

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

       1. clear-num 77.7%

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

       2. un-div-inv 77.7%

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

   12. Applied egg-rr 77.7%

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

       1. associate-/r/ 80.1%

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

   14. Simplified 80.1%

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

   if -2.4e6 < y.re < 3.74999999999999984e-47

   1. Initial program 69.6%

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

      1. fma-define 69.6%

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

      2. fma-define 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in y.im around inf 85.6%

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

      1. *-commutative 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. times-frac 85.7%

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

   7. Applied egg-rr 85.7%

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

   if 3.74999999999999984e-47 < y.re

   1. Initial program 47.6%

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

      1. fma-define 47.6%

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

      2. fma-define 47.6%

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

   3. Simplified 47.6%

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

      1. *-un-lft-identity 47.6%

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

      2. fma-define 47.6%

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

      3. add-sqr-sqrt 47.6%

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

      4. times-frac 47.5%

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

      5. fma-define 47.5%

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

      6. hypot-define 47.6%

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

      7. fma-define 47.6%

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

      8. fma-define 47.6%

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

      9. hypot-define 73.8%

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

   6. Applied egg-rr 73.8%

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

   7. Taylor expanded in y.re around inf 68.3%

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

      1. associate-/l* 74.1%

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

   9. Simplified 74.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2400000:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 8.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.5e-18)
   (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
   (if (<= y.im 8.4e-150)
     (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
     (/ (+ x.im (* y.re (/ x.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 <= -5.5e-18) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 8.4e-150) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5.5d-18)) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else if (y_46im <= 8.4d-150) then
        tmp = (x_46re + ((x_46im * y_46im) / y_46re)) / y_46re
    else
        tmp = (x_46im + (y_46re * (x_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.5e-18) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 8.4e-150) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.5e-18:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	elif y_46_im <= 8.4e-150:
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = (x_46_im + (y_46_re * (x_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 <= -5.5e-18)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	elseif (y_46_im <= 8.4e-150)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.5e-18)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	elseif (y_46_im <= 8.4e-150)
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	else
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.5e-18], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 8.4e-150], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.5e-18

   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. Step-by-step derivation

      1. fma-define 49.1%

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

      2. fma-define 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in y.im around inf 71.3%

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

      1. associate-/l* 74.4%

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

   7. Simplified 74.4%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.5%

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

   9. Applied egg-rr 74.5%

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

   if -5.5e-18 < y.im < 8.4000000000000004e-150

   1. Initial program 70.5%

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

      1. fma-define 70.5%

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

      2. fma-define 70.5%

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

   3. Simplified 70.5%

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

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

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

      1. *-commutative 88.0%

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

   7. Simplified 88.0%

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

   if 8.4000000000000004e-150 < 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. Step-by-step derivation

      1. fma-define 49.1%

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

      2. fma-define 49.1%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in y.im around inf 76.1%

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

      1. *-commutative 76.1%

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

      2. *-un-lft-identity 76.1%

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

      3. times-frac 77.4%

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

   7. Applied egg-rr 77.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 8.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-18} \lor \neg \left(y.im \leq 3.1 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3e-18) (not (<= y.im 3.1e-156)))
   (/ 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 <= -3e-18) || !(y_46_im <= 3.1e-156)) {
		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 <= (-3d-18)) .or. (.not. (y_46im <= 3.1d-156))) 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 <= -3e-18) || !(y_46_im <= 3.1e-156)) {
		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 <= -3e-18) or not (y_46_im <= 3.1e-156):
		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 <= -3e-18) || !(y_46_im <= 3.1e-156))
		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 <= -3e-18) || ~((y_46_im <= 3.1e-156)))
		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, -3e-18], N[Not[LessEqual[y$46$im, 3.1e-156]], $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 < -2.99999999999999983e-18 or 3.0999999999999998e-156 < y.im

   1. Initial program 49.7%

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

      1. fma-define 49.7%

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

      2. fma-define 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in y.re around 0 64.9%

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

   if -2.99999999999999983e-18 < y.im < 3.0999999999999998e-156

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in y.re around inf 69.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-18} \lor \neg \left(y.im \leq 3.1 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.8%

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

   1. fma-define 57.8%

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

   2. fma-define 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in y.re around 0 43.8%

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

Reproduce

herbie shell --seed 2024141 
(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))))