_divideComplex, real part

Percentage Accurate: 61.7% → 85.9%

Time: 13.3s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 85.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+295)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (/ (+ 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 ((((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+295) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_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 (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+295)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / 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+295], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $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))) < 2e295

   1. Initial program 78.6%

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

      1. fma-define 78.6%

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

      2. fma-define 78.7%

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

   3. Simplified 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. fma-define 78.6%

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

      3. add-sqr-sqrt 78.6%

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

      4. times-frac 78.6%

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

      5. fma-define 78.6%

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

      6. hypot-define 78.6%

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

      7. fma-define 78.6%

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

      8. fma-define 78.6%

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

      9. hypot-define 97.6%

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

   6. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*l/ 97.7%

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

      3. div-inv 97.8%

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

      4. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 2e295 < (/.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 12.0%

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

      1. fma-define 12.0%

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

      2. fma-define 12.0%

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

   3. Simplified 12.0%

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

   5. Taylor expanded in y.im around inf 49.0%

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

      1. associate-/l* 58.6%

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

   7. Simplified 58.6%

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

      1. clear-num 58.6%

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

      2. un-div-inv 58.7%

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

   9. Applied egg-rr 58.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+65}:\\ \;\;\;\;\left(x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{1}{y.re} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.92 \cdot 10^{-149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 105:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot \left(1 + x.im \cdot \frac{y.im}{x.re \cdot y.re}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1e+65)
   (+ (* (* x.im (/ y.im y.re)) (/ 1.0 y.re)) (/ x.re y.re))
   (if (<= y.re -1.92e-149)
     (/ (fma y.re x.re (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 105.0)
       (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
       (/ (* x.re (+ 1.0 (* x.im (/ y.im (* x.re 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 <= -1e+65) {
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.92e-149) {
		tmp = fma(y_46_re, x_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 <= 105.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= -1e+65)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) * Float64(1.0 / y_46_re)) + Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.92e-149)
		tmp = Float64(fma(y_46_re, x_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 <= 105.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re * Float64(1.0 + Float64(x_46_im * Float64(y_46_im / Float64(x_46_re * y_46_re))))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1e+65], N[(N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.92e-149], N[(N[(y$46$re * x$46$re + 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, 105.0], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re * N[(1.0 + N[(x$46$im * N[(y$46$im / N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.9999999999999999e64

   1. Initial program 43.3%

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

      1. fma-define 43.3%

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

      2. fma-define 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in y.re around inf 75.7%

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

      1. *-commutative 75.7%

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

   7. Simplified 75.7%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. +-commutative 74.6%

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

      4. associate-/l* 78.9%

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

      5. fma-define 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. unpow-1 78.9%

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

       2. fma-undefine 78.9%

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

       3. *-commutative 78.9%

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

       4. associate-*l/ 74.6%

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

       5. associate-*r/ 78.9%

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

       6. fma-define 78.9%

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

   11. Simplified 78.9%

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

       1. associate-/r/ 79.7%

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

       2. fma-undefine 79.7%

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

       3. distribute-rgt-in 79.7%

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

       4. div-inv 79.9%

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

   13. Applied egg-rr 79.9%

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

   if -9.9999999999999999e64 < y.re < -1.92e-149

   1. Initial program 87.5%

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

      1. *-commutative 87.5%

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

      2. fma-define 87.5%

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

   4. Applied egg-rr 87.5%

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

   if -1.92e-149 < y.re < 105

   1. Initial program 68.6%

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

      1. fma-define 68.6%

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

      2. fma-define 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in y.im around inf 91.6%

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

      1. associate-/l* 91.6%

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

   7. Simplified 91.6%

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

      1. clear-num 91.6%

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

      2. un-div-inv 91.7%

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

   9. Applied egg-rr 91.7%

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

   if 105 < y.re

   1. Initial program 43.4%

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

      1. fma-define 43.4%

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

      2. fma-define 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x.re around inf 77.1%

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

      1. associate-/l* 83.9%

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

   10. Simplified 83.9%

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

Alternative 3: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+64}:\\ \;\;\;\;\left(x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{1}{y.re} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.5:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot \left(1 + x.im \cdot \frac{y.im}{x.re \cdot y.re}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9.2e+64)
   (+ (* (* x.im (/ y.im y.re)) (/ 1.0 y.re)) (/ x.re y.re))
   (if (<= y.re -1.8e-147)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 9.5)
       (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
       (/ (* x.re (+ 1.0 (* x.im (/ y.im (* x.re 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 <= -9.2e+64) {
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.8e-147) {
		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 <= 9.5) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= (-9.2d+64)) then
        tmp = ((x_46im * (y_46im / y_46re)) * (1.0d0 / y_46re)) + (x_46re / y_46re)
    else if (y_46re <= (-1.8d-147)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 9.5d0) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = (x_46re * (1.0d0 + (x_46im * (y_46im / (x_46re * 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 <= -9.2e+64) {
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.8e-147) {
		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 <= 9.5) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= -9.2e+64:
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re)
	elif y_46_re <= -1.8e-147:
		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 <= 9.5:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	else:
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= -9.2e+64)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) * Float64(1.0 / y_46_re)) + Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.8e-147)
		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 <= 9.5)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re * Float64(1.0 + Float64(x_46_im * Float64(y_46_im / Float64(x_46_re * 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 <= -9.2e+64)
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	elseif (y_46_re <= -1.8e-147)
		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 <= 9.5)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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, -9.2e+64], N[(N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.8e-147], 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, 9.5], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re * N[(1.0 + N[(x$46$im * N[(y$46$im / N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.2e64

   1. Initial program 43.3%

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

      1. fma-define 43.3%

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

      2. fma-define 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in y.re around inf 75.7%

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

      1. *-commutative 75.7%

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

   7. Simplified 75.7%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. +-commutative 74.6%

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

      4. associate-/l* 78.9%

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

      5. fma-define 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. unpow-1 78.9%

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

       2. fma-undefine 78.9%

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

       3. *-commutative 78.9%

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

       4. associate-*l/ 74.6%

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

       5. associate-*r/ 78.9%

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

       6. fma-define 78.9%

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

   11. Simplified 78.9%

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

       1. associate-/r/ 79.7%

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

       2. fma-undefine 79.7%

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

       3. distribute-rgt-in 79.7%

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

       4. div-inv 79.9%

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

   13. Applied egg-rr 79.9%

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

   if -9.2e64 < y.re < -1.80000000000000006e-147

   1. Initial program 87.5%

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

   if -1.80000000000000006e-147 < y.re < 9.5

   1. Initial program 68.6%

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

      1. fma-define 68.6%

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

      2. fma-define 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in y.im around inf 91.6%

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

      1. associate-/l* 91.6%

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

   7. Simplified 91.6%

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

      1. clear-num 91.6%

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

      2. un-div-inv 91.7%

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

   9. Applied egg-rr 91.7%

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

   if 9.5 < y.re

   1. Initial program 43.4%

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

      1. fma-define 43.4%

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

      2. fma-define 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x.re around inf 77.1%

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

      1. associate-/l* 83.9%

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

   10. Simplified 83.9%

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

Alternative 4: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{+63}:\\ \;\;\;\;\left(x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{1}{y.re} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 65:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot \left(1 + x.im \cdot \frac{y.im}{x.re \cdot y.re}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.7e+63)
   (+ (* (* x.im (/ y.im y.re)) (/ 1.0 y.re)) (/ x.re y.re))
   (if (<= y.re -1.3e-41)
     (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 65.0)
       (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
       (/ (* x.re (+ 1.0 (* x.im (/ y.im (* x.re 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 <= -4.7e+63) {
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.3e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 65.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= (-4.7d+63)) then
        tmp = ((x_46im * (y_46im / y_46re)) * (1.0d0 / y_46re)) + (x_46re / y_46re)
    else if (y_46re <= (-1.3d-41)) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 65.0d0) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = (x_46re * (1.0d0 + (x_46im * (y_46im / (x_46re * 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 <= -4.7e+63) {
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.3e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 65.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= -4.7e+63:
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re)
	elif y_46_re <= -1.3e-41:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 65.0:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	else:
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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 <= -4.7e+63)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) * Float64(1.0 / y_46_re)) + Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.3e-41)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 65.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re * Float64(1.0 + Float64(x_46_im * Float64(y_46_im / Float64(x_46_re * 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 <= -4.7e+63)
		tmp = ((x_46_im * (y_46_im / y_46_re)) * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	elseif (y_46_re <= -1.3e-41)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 65.0)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = (x_46_re * (1.0 + (x_46_im * (y_46_im / (x_46_re * 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, -4.7e+63], N[(N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.3e-41], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 65.0], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re * N[(1.0 + N[(x$46$im * N[(y$46$im / N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.7000000000000003e63

   1. Initial program 43.3%

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

      1. fma-define 43.3%

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

      2. fma-define 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in y.re around inf 75.7%

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

      1. *-commutative 75.7%

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

   7. Simplified 75.7%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. +-commutative 74.6%

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

      4. associate-/l* 78.9%

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

      5. fma-define 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. unpow-1 78.9%

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

       2. fma-undefine 78.9%

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

       3. *-commutative 78.9%

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

       4. associate-*l/ 74.6%

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

       5. associate-*r/ 78.9%

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

       6. fma-define 78.9%

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

   11. Simplified 78.9%

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

       1. associate-/r/ 79.7%

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

       2. fma-undefine 79.7%

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

       3. distribute-rgt-in 79.7%

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

       4. div-inv 79.9%

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

   13. Applied egg-rr 79.9%

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

   if -4.7000000000000003e63 < y.re < -1.3e-41

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. fma-define 92.1%

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

   4. Applied egg-rr 92.1%

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

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

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

   if -1.3e-41 < y.re < 65

   1. Initial program 71.3%

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

      1. fma-define 71.3%

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

      2. fma-define 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in y.im around inf 85.7%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. clear-num 85.7%

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

      2. un-div-inv 85.8%

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

   9. Applied egg-rr 85.8%

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

   if 65 < y.re

   1. Initial program 43.4%

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

      1. fma-define 43.4%

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

      2. fma-define 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x.re around inf 77.1%

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

      1. associate-/l* 83.9%

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

   10. Simplified 83.9%

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

Alternative 5: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.95 \cdot 10^{+63}:\\ \;\;\;\;t\_0 \cdot \frac{1}{y.re} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 23000:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t\_0}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.im y.re))))
   (if (<= y.re -2.95e+63)
     (+ (* t_0 (/ 1.0 y.re)) (/ x.re y.re))
     (if (<= y.re -1.05e-41)
       (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 23000.0)
         (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
         (/ (+ x.re t_0) 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_im * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.95e+63) {
		tmp = (t_0 * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.05e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 23000.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re + t_0) / 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_46im * (y_46im / y_46re)
    if (y_46re <= (-2.95d+63)) then
        tmp = (t_0 * (1.0d0 / y_46re)) + (x_46re / y_46re)
    else if (y_46re <= (-1.05d-41)) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 23000.0d0) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = (x_46re + t_0) / 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_im * (y_46_im / y_46_re);
	double tmp;
	if (y_46_re <= -2.95e+63) {
		tmp = (t_0 * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.05e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 23000.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = (x_46_re + t_0) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_im * (y_46_im / y_46_re)
	tmp = 0
	if y_46_re <= -2.95e+63:
		tmp = (t_0 * (1.0 / y_46_re)) + (x_46_re / y_46_re)
	elif y_46_re <= -1.05e-41:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 23000.0:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	else:
		tmp = (x_46_re + t_0) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_im * Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.95e+63)
		tmp = Float64(Float64(t_0 * Float64(1.0 / y_46_re)) + Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.05e-41)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 23000.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + t_0) / 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_im * (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.95e+63)
		tmp = (t_0 * (1.0 / y_46_re)) + (x_46_re / y_46_re);
	elseif (y_46_re <= -1.05e-41)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 23000.0)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = (x_46_re + t_0) / 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[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.95e+63], N[(N[(t$95$0 * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.05e-41], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 23000.0], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + t$95$0), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.95000000000000014e63

   1. Initial program 43.3%

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

      1. fma-define 43.3%

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

      2. fma-define 43.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in y.re around inf 75.7%

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

      1. *-commutative 75.7%

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

   7. Simplified 75.7%

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

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. +-commutative 74.6%

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

      4. associate-/l* 78.9%

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

      5. fma-define 78.9%

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

   9. Applied egg-rr 78.9%

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

       1. unpow-1 78.9%

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

       2. fma-undefine 78.9%

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

       3. *-commutative 78.9%

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

       4. associate-*l/ 74.6%

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

       5. associate-*r/ 78.9%

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

       6. fma-define 78.9%

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

   11. Simplified 78.9%

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

       1. associate-/r/ 79.7%

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

       2. fma-undefine 79.7%

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

       3. distribute-rgt-in 79.7%

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

       4. div-inv 79.9%

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

   13. Applied egg-rr 79.9%

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

   if -2.95000000000000014e63 < y.re < -1.05000000000000006e-41

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. fma-define 92.1%

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

   4. Applied egg-rr 92.1%

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

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

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

   if -1.05000000000000006e-41 < y.re < 23000

   1. Initial program 71.3%

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

      1. fma-define 71.3%

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

      2. fma-define 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in y.im around inf 85.7%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. clear-num 85.7%

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

      2. un-div-inv 85.8%

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

   9. Applied egg-rr 85.8%

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

   if 23000 < y.re

   1. Initial program 43.4%

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

      1. fma-define 43.4%

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

      2. fma-define 43.4%

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

   3. Simplified 43.4%

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

   5. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in y.im around 0 77.2%

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

      1. associate-*r/ 82.7%

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

   10. Simplified 82.7%

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

Alternative 6: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 920:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))
   (if (<= y.re -8e+63)
     t_0
     (if (<= y.re -1.3e-41)
       (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 920.0) (/ (+ x.im (/ x.re (/ y.im y.re))) y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8e+63) {
		tmp = t_0;
	} else if (y_46_re <= -1.3e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 920.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-8d+63)) then
        tmp = t_0
    else if (y_46re <= (-1.3d-41)) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 920.0d0) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -8e+63) {
		tmp = t_0;
	} else if (y_46_re <= -1.3e-41) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 920.0) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -8e+63:
		tmp = t_0
	elif y_46_re <= -1.3e-41:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 920.0:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -8e+63)
		tmp = t_0;
	elseif (y_46_re <= -1.3e-41)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 920.0)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -8e+63)
		tmp = t_0;
	elseif (y_46_re <= -1.3e-41)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 920.0)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -8e+63], t$95$0, If[LessEqual[y$46$re, -1.3e-41], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 920.0], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.00000000000000046e63 or 920 < y.re

   1. Initial program 43.4%

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

      1. fma-define 43.4%

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

      2. fma-define 43.4%

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

   3. Simplified 43.4%

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

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

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

      1. *-commutative 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in y.im around 0 76.6%

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

      1. associate-*r/ 81.6%

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

   10. Simplified 81.6%

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

   if -8.00000000000000046e63 < y.re < -1.3e-41

   1. Initial program 92.1%

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

      1. *-commutative 92.1%

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

      2. fma-define 92.1%

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

   4. Applied egg-rr 92.1%

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

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

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

   if -1.3e-41 < y.re < 920

   1. Initial program 71.3%

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

      1. fma-define 71.3%

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

      2. fma-define 71.3%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in y.im around inf 85.7%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. clear-num 85.7%

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

      2. un-div-inv 85.8%

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

   9. Applied egg-rr 85.8%

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

Alternative 7: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+61} \lor \neg \left(y.re \leq 210\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -7.5e+61) (not (<= y.re 210.0)))
   (/ 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 <= -7.5e+61) || !(y_46_re <= 210.0)) {
		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 <= (-7.5d+61)) .or. (.not. (y_46re <= 210.0d0))) 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 <= -7.5e+61) || !(y_46_re <= 210.0)) {
		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 <= -7.5e+61) or not (y_46_re <= 210.0):
		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 <= -7.5e+61) || !(y_46_re <= 210.0))
		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 <= -7.5e+61) || ~((y_46_re <= 210.0)))
		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, -7.5e+61], N[Not[LessEqual[y$46$re, 210.0]], $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 < -7.5e61 or 210 < y.re

   1. Initial program 43.8%

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

      1. fma-define 43.8%

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

      2. fma-define 43.8%

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

   3. Simplified 43.8%

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

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

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

   if -7.5e61 < y.re < 210

   1. Initial program 75.0%

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

      1. fma-define 75.1%

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

      2. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in y.im around inf 79.3%

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

      1. associate-/l* 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num 80.0%

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

      2. un-div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+61} \lor \neg \left(y.re \leq 210\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+62} \lor \neg \left(y.re \leq 250\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.5e+62) (not (<= y.re 250.0)))
   (/ 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 <= -2.5e+62) || !(y_46_re <= 250.0)) {
		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 <= (-2.5d+62)) .or. (.not. (y_46re <= 250.0d0))) 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 <= -2.5e+62) || !(y_46_re <= 250.0)) {
		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 <= -2.5e+62) or not (y_46_re <= 250.0):
		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 <= -2.5e+62) || !(y_46_re <= 250.0))
		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 <= -2.5e+62) || ~((y_46_re <= 250.0)))
		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, -2.5e+62], N[Not[LessEqual[y$46$re, 250.0]], $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 < -2.50000000000000014e62 or 250 < y.re

   1. Initial program 43.8%

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

      1. fma-define 43.8%

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

      2. fma-define 43.8%

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

   3. Simplified 43.8%

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

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

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

   if -2.50000000000000014e62 < y.re < 250

   1. Initial program 75.0%

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

      1. fma-define 75.1%

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

      2. fma-define 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in y.im around inf 79.3%

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

      1. associate-/l* 80.0%

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

   7. Simplified 80.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+62} \lor \neg \left(y.re \leq 250\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 1050000000:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.55e-25)
   (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
   (if (<= y.im 1050000000.0)
     (/ (+ 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_im <= -1.55e-25) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 1050000000.0) {
		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_46im <= (-1.55d-25)) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else if (y_46im <= 1050000000.0d0) 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_im <= -1.55e-25) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 1050000000.0) {
		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_im <= -1.55e-25:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	elif y_46_im <= 1050000000.0:
		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_im <= -1.55e-25)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	elseif (y_46_im <= 1050000000.0)
		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(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_im <= -1.55e-25)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	elseif (y_46_im <= 1050000000.0)
		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[LessEqual[y$46$im, -1.55e-25], 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, 1050000000.0], 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[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.54999999999999997e-25

   1. Initial program 55.7%

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

      1. fma-define 55.7%

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

      2. fma-define 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in y.im around inf 72.7%

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

      1. associate-/l* 76.5%

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

   7. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

   9. Applied egg-rr 76.5%

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

   if -1.54999999999999997e-25 < y.im < 1.05e9

   1. Initial program 73.9%

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

      1. fma-define 73.9%

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

      2. fma-define 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in y.re around inf 87.9%

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

      1. *-commutative 87.9%

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

   7. Simplified 87.9%

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

   if 1.05e9 < y.im

   1. Initial program 39.8%

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

      1. fma-define 39.8%

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

      2. fma-define 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in y.im around inf 68.3%

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

      1. associate-/l* 74.8%

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

   7. Simplified 74.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 1050000000:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 150000000:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3e-25)
   (/ (+ x.im (/ x.re (/ y.im y.re))) y.im)
   (if (<= y.im 150000000.0)
     (/ (+ 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_im <= -3e-25) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 150000000.0) {
		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_46im <= (-3d-25)) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / y_46im
    else if (y_46im <= 150000000.0d0) 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_im <= -3e-25) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	} else if (y_46_im <= 150000000.0) {
		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_im <= -3e-25:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im
	elif y_46_im <= 150000000.0:
		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_im <= -3e-25)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / y_46_im);
	elseif (y_46_im <= 150000000.0)
		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(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_im <= -3e-25)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / y_46_im;
	elseif (y_46_im <= 150000000.0)
		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[LessEqual[y$46$im, -3e-25], 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, 150000000.0], 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[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.9999999999999998e-25

   1. Initial program 55.7%

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

      1. fma-define 55.7%

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

      2. fma-define 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in y.im around inf 72.7%

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

      1. associate-/l* 76.5%

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

   7. Simplified 76.5%

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

      1. clear-num 76.5%

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

      2. un-div-inv 76.5%

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

   9. Applied egg-rr 76.5%

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

   if -2.9999999999999998e-25 < y.im < 1.5e8

   1. Initial program 73.9%

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

      1. fma-define 73.9%

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

      2. fma-define 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in y.re around inf 87.9%

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

      1. *-commutative 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in y.im around 0 87.9%

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

      1. associate-*r/ 87.2%

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

   10. Simplified 87.2%

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

   if 1.5e8 < y.im

   1. Initial program 39.8%

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

      1. fma-define 39.8%

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

      2. fma-define 39.8%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in y.im around inf 68.3%

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

      1. associate-/l* 74.8%

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

   7. Simplified 74.8%

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

Alternative 11: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{-13} \lor \neg \left(y.im \leq 1200000000000\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.6e-13) (not (<= y.im 1200000000000.0)))
   (/ 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 <= -2.6e-13) || !(y_46_im <= 1200000000000.0)) {
		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 <= (-2.6d-13)) .or. (.not. (y_46im <= 1200000000000.0d0))) 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 <= -2.6e-13) || !(y_46_im <= 1200000000000.0)) {
		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 <= -2.6e-13) or not (y_46_im <= 1200000000000.0):
		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 <= -2.6e-13) || !(y_46_im <= 1200000000000.0))
		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 <= -2.6e-13) || ~((y_46_im <= 1200000000000.0)))
		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, -2.6e-13], N[Not[LessEqual[y$46$im, 1200000000000.0]], $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 < -2.6e-13 or 1.2e12 < y.im

   1. Initial program 47.2%

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

      1. fma-define 47.2%

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

      2. fma-define 47.2%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in y.re around 0 66.3%

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

   if -2.6e-13 < y.im < 1.2e12

   1. Initial program 74.7%

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

      1. fma-define 74.7%

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

      2. fma-define 74.7%

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

   3. Simplified 74.7%

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

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

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{-13} \lor \neg \left(y.im \leq 1200000000000\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. fma-define 60.4%

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

   2. fma-define 60.4%

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

3. Simplified 60.4%

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

5. Taylor expanded in y.re around 0 42.0%

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

Reproduce

herbie shell --seed 2024172 
(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))))