_divideComplex, real part

Percentage Accurate: 61.8% → 80.8%

Time: 14.5s

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: 61.8% 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: 80.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ t_1 := \sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+57}:\\ \;\;\;\;\frac{{t\_1}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im))
        (t_1 (cbrt (fma x.re y.re (* y.im x.im)))))
   (if (<= y.im -7.5e+25)
     t_0
     (if (<= y.im 1.45e-74)
       (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
       (if (<= y.im 1.02e+57)
         (* (/ (pow t_1 2.0) (hypot y.re y.im)) (/ t_1 (hypot 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_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double t_1 = cbrt(fma(x_46_re, y_46_re, (y_46_im * x_46_im)));
	double tmp;
	if (y_46_im <= -7.5e+25) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e-74) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.02e+57) {
		tmp = (pow(t_1, 2.0) / hypot(y_46_re, y_46_im)) * (t_1 / hypot(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_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	t_1 = cbrt(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)))
	tmp = 0.0
	if (y_46_im <= -7.5e+25)
		tmp = t_0;
	elseif (y_46_im <= 1.45e-74)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.02e+57)
		tmp = Float64(Float64((t_1 ^ 2.0) / hypot(y_46_re, y_46_im)) * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e+25], t$95$0, If[LessEqual[y$46$im, 1.45e-74], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.02e+57], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.49999999999999993e25 or 1.02e57 < y.im

   1. Initial program 53.2%

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

      1. fma-define 53.2%

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

      2. fma-define 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in y.im around inf 80.3%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

   if -7.49999999999999993e25 < y.im < 1.45e-74

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

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

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 1.45e-74 < y.im < 1.02e57

   1. Initial program 80.6%

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

      1. fma-define 80.6%

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

      2. fma-define 80.7%

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

   3. Simplified 80.7%

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

      1. add-cube-cbrt 79.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

      2. add-sqr-sqrt 79.6%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. times-frac 79.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      4. pow2 79.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      5. fma-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      6. hypot-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. fma-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      8. hypot-define 98.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Applied egg-rr 98.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+57}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -2.7e+29)
     t_0
     (if (<= y.im 2.95e-77)
       (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
       (if (<= y.im 3.3e+59)
         (/
          1.0
          (/
           (hypot y.re y.im)
           (/ (fma x.re y.re (* y.im x.im)) (hypot 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_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.7e+29) {
		tmp = t_0;
	} else if (y_46_im <= 2.95e-77) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.3e+59) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(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_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.7e+29)
		tmp = t_0;
	elseif (y_46_im <= 2.95e-77)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.3e+59)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.7e+29], t$95$0, If[LessEqual[y$46$im, 2.95e-77], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.3e+59], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.7e29 or 3.2999999999999999e59 < y.im

   1. Initial program 52.8%

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

      1. fma-define 52.8%

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

      2. fma-define 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y.im around inf 80.1%

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

      1. associate-/l* 84.1%

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

   7. Simplified 84.1%

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

   if -2.7e29 < y.im < 2.94999999999999982e-77

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

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

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 2.94999999999999982e-77 < y.im < 3.2999999999999999e59

   1. Initial program 81.3%

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

      1. fma-define 81.3%

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

      2. fma-define 81.3%

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

   3. Simplified 81.3%

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

      1. add-cube-cbrt 80.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

      2. add-sqr-sqrt 80.2%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      4. pow2 80.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      5. fma-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      6. hypot-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. fma-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      8. hypot-define 98.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.0%

         \[\leadsto \color{blue}{\frac{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      2. clear-num 95.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}} \]

      3. associate-*l/ 95.3%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}}} \]

      4. unpow2 95.3%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}} \]

      5. add-cube-cbrt 97.0%

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

   8. Applied egg-rr 97.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -4.5e+26)
     t_0
     (if (<= y.im 1.2e-76)
       (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
       (if (<= y.im 3.3e+59)
         (/
          1.0
          (*
           (hypot y.re y.im)
           (/ (hypot y.re y.im) (fma y.re x.re (* y.im x.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_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -4.5e+26) {
		tmp = t_0;
	} else if (y_46_im <= 1.2e-76) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.3e+59) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / fma(y_46_re, x_46_re, (y_46_im * x_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_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.5e+26)
		tmp = t_0;
	elseif (y_46_im <= 1.2e-76)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.3e+59)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(y_46_re, y_46_im) / fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.5e+26], t$95$0, If[LessEqual[y$46$im, 1.2e-76], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.3e+59], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.49999999999999978e26 or 3.2999999999999999e59 < y.im

   1. Initial program 52.8%

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

      1. fma-define 52.8%

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

      2. fma-define 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y.im around inf 80.1%

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

      1. associate-/l* 84.1%

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

   7. Simplified 84.1%

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

   if -4.49999999999999978e26 < y.im < 1.20000000000000007e-76

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

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

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 1.20000000000000007e-76 < y.im < 3.2999999999999999e59

   1. Initial program 81.3%

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

      1. fma-define 81.3%

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

      2. fma-define 81.3%

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

   3. Simplified 81.3%

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

      1. add-cube-cbrt 80.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

      2. add-sqr-sqrt 80.2%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. times-frac 80.1%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      4. pow2 80.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      5. fma-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      6. hypot-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. fma-define 80.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      8. hypot-define 98.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Applied egg-rr 98.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.0%

         \[\leadsto \color{blue}{\frac{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      2. clear-num 95.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}} \]

      3. associate-*l/ 95.3%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}}} \]

      4. unpow2 95.3%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}} \]

      5. add-cube-cbrt 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. div-inv 96.8%

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

      2. clear-num 96.8%

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

      3. fma-undefine 96.8%

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

      4. *-commutative 96.8%

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

      5. fma-define 96.8%

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

      6. *-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

Alternative 4: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -6.8e+25)
     t_0
     (if (<= y.im 7.7e-75)
       (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
       (if (<= y.im 3.2e+59)
         (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im 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_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -6.8e+25) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e-75) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.2e+59) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * 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_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.8e+25)
		tmp = t_0;
	elseif (y_46_im <= 7.7e-75)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.2e+59)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6.8e+25], t$95$0, If[LessEqual[y$46$im, 7.7e-75], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.2e+59], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.79999999999999967e25 or 3.19999999999999982e59 < y.im

   1. Initial program 52.8%

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

      1. fma-define 52.8%

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

      2. fma-define 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y.im around inf 80.1%

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

      1. associate-/l* 84.1%

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

   7. Simplified 84.1%

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

   if -6.79999999999999967e25 < y.im < 7.69999999999999958e-75

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

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

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 7.69999999999999958e-75 < y.im < 3.19999999999999982e59

   1. Initial program 81.3%

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

      1. fma-define 81.3%

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

      2. fma-define 81.3%

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

   3. Simplified 81.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+25)
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (if (<= y.im 1.3e-53)
     (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
     (/ (fma x.re (/ y.re y.im) x.im) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -9e+25) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= 1.3e-53) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -9e+25)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= 1.3e-53)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -9e+25], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e-53], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.0000000000000006e25

   1. Initial program 55.1%

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

      1. fma-define 55.1%

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

      2. fma-define 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in y.im around inf 80.9%

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

      1. associate-/l* 86.4%

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

   7. Simplified 86.4%

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

   if -9.0000000000000006e25 < y.im < 1.29999999999999998e-53

   1. Initial program 69.0%

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

      1. fma-define 69.1%

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

      2. fma-define 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in y.re around inf 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

   if 1.29999999999999998e-53 < y.im

   1. Initial program 61.2%

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

      1. fma-define 61.2%

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

      2. fma-define 61.2%

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

   3. Simplified 61.2%

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

   5. Taylor expanded in y.im around inf 76.2%

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

      1. +-commutative 76.2%

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

      2. associate-/l* 77.6%

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

      3. fma-define 77.6%

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

   7. Simplified 77.6%

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

Alternative 6: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.12 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -9.5e+25)
     t_0
     (if (<= y.im 2.12e-69)
       (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
       (if (<= y.im 2.8e+59)
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.im y.im) (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -9.5e+25) {
		tmp = t_0;
	} else if (y_46_im <= 2.12e-69) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.8e+59) {
		tmp = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-9.5d+25)) then
        tmp = t_0
    else if (y_46im <= 2.12d-69) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else if (y_46im <= 2.8d+59) then
        tmp = ((y_46im * x_46im) + (x_46re * y_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -9.5e+25) {
		tmp = t_0;
	} else if (y_46_im <= 2.12e-69) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.8e+59) {
		tmp = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -9.5e+25:
		tmp = t_0
	elif y_46_im <= 2.12e-69:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	elif y_46_im <= 2.8e+59:
		tmp = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.5e+25)
		tmp = t_0;
	elseif (y_46_im <= 2.12e-69)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.8e+59)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -9.5e+25)
		tmp = t_0;
	elseif (y_46_im <= 2.12e-69)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.8e+59)
		tmp = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+25], t$95$0, If[LessEqual[y$46$im, 2.12e-69], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+59], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.5000000000000005e25 or 2.7999999999999998e59 < y.im

   1. Initial program 52.8%

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

      1. fma-define 52.8%

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

      2. fma-define 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in y.im around inf 80.1%

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

      1. associate-/l* 84.1%

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

   7. Simplified 84.1%

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

   if -9.5000000000000005e25 < y.im < 2.11999999999999993e-69

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

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

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 2.11999999999999993e-69 < y.im < 2.7999999999999998e59

   1. Initial program 81.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.12 \cdot 10^{-69}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.2e+26) (not (<= y.im 1.3e-53)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e+26) || !(y_46_im <= 1.3e-53)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-8.2d+26)) .or. (.not. (y_46im <= 1.3d-53))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.2e+26) || !(y_46_im <= 1.3e-53)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.2e+26) or not (y_46_im <= 1.3e-53):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.2e+26) || !(y_46_im <= 1.3e-53))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -8.2e+26) || ~((y_46_im <= 1.3e-53)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -8.2e+26], N[Not[LessEqual[y$46$im, 1.3e-53]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.19999999999999967e26 or 1.29999999999999998e-53 < y.im

   1. Initial program 58.6%

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

      1. fma-define 58.6%

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

      2. fma-define 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in y.im around inf 78.2%

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

      1. associate-/l* 81.4%

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

   7. Simplified 81.4%

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

   if -8.19999999999999967e26 < y.im < 1.29999999999999998e-53

   1. Initial program 69.0%

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

      1. fma-define 69.1%

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

      2. fma-define 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in y.re around inf 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.06 \cdot 10^{+29} \lor \neg \left(y.im \leq 10^{-53}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.06e+29) (not (<= y.im 1e-53)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.06e+29) || !(y_46_im <= 1e-53)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.06d+29)) .or. (.not. (y_46im <= 1d-53))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.06e+29) || !(y_46_im <= 1e-53)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.06e+29) or not (y_46_im <= 1e-53):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.06e+29) || !(y_46_im <= 1e-53))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.06e+29) || ~((y_46_im <= 1e-53)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.06e+29], N[Not[LessEqual[y$46$im, 1e-53]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.0600000000000001e29 or 1.00000000000000003e-53 < y.im

   1. Initial program 58.6%

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

      1. fma-define 58.6%

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

      2. fma-define 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in y.im around inf 78.2%

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

      1. associate-/l* 81.4%

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

   7. Simplified 81.4%

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

   if -1.0600000000000001e29 < y.im < 1.00000000000000003e-53

   1. Initial program 69.0%

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

      1. fma-define 69.1%

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

      2. fma-define 69.1%

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

   3. Simplified 69.1%

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

      1. add-cube-cbrt 68.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

      2. add-sqr-sqrt 68.4%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. times-frac 68.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      4. pow2 68.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      5. fma-define 68.3%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      6. hypot-define 68.4%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. fma-define 68.4%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      8. hypot-define 79.6%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   7. Taylor expanded in y.re around inf 90.3%

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

      1. associate-/l* 89.1%

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

   9. Simplified 89.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.06 \cdot 10^{+29} \lor \neg \left(y.im \leq 10^{-53}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+15} \lor \neg \left(y.im \leq 3.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -8.5e+15) (not (<= y.im 3.7e-59)))
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.5e+15) || !(y_46_im <= 3.7e-59)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-8.5d+15)) .or. (.not. (y_46im <= 3.7d-59))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -8.5e+15) || !(y_46_im <= 3.7e-59)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -8.5e+15) or not (y_46_im <= 3.7e-59):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8.5e+15) || !(y_46_im <= 3.7e-59))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -8.5e+15) || ~((y_46_im <= 3.7e-59)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -8.5e+15], N[Not[LessEqual[y$46$im, 3.7e-59]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.5e15 or 3.6999999999999999e-59 < y.im

   1. Initial program 59.0%

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

      1. fma-define 59.0%

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

      2. fma-define 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y.im around inf 76.8%

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

      1. associate-/l* 79.9%

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

   7. Simplified 79.9%

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

   if -8.5e15 < y.im < 3.6999999999999999e-59

   1. Initial program 68.8%

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

      1. fma-define 68.8%

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

      2. fma-define 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in y.re around inf 69.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+15} \lor \neg \left(y.im \leq 3.7 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+22} \lor \neg \left(y.im \leq 1.2 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.6e+22) (not (<= y.im 1.2e-53)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.6e+22) || !(y_46_im <= 1.2e-53)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.6d+22)) .or. (.not. (y_46im <= 1.2d-53))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.6e+22) || !(y_46_im <= 1.2e-53)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.6e+22) or not (y_46_im <= 1.2e-53):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.6e+22) || !(y_46_im <= 1.2e-53))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.6e+22) || ~((y_46_im <= 1.2e-53)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.6e+22], N[Not[LessEqual[y$46$im, 1.2e-53]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.6e22 or 1.20000000000000004e-53 < y.im

   1. Initial program 59.1%

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

      1. fma-define 59.1%

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

      2. fma-define 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y.re around 0 63.9%

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

   if -1.6e22 < y.im < 1.20000000000000004e-53

   1. Initial program 68.5%

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

      1. fma-define 68.5%

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

      2. fma-define 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in y.re around inf 68.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+22} \lor \neg \left(y.im \leq 1.2 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 1.55 \cdot 10^{+197}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 1.55e+197) (/ x.im y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 1.55e+197) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= 1.55d+197) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 1.55e+197) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 1.55e+197:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 1.55e+197)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 1.55e+197)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 1.55e+197], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.55e197

   1. Initial program 65.7%

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

      1. fma-define 65.7%

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

      2. fma-define 65.7%

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

   3. Simplified 65.7%

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

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

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

   if 1.55e197 < y.re

   1. Initial program 32.4%

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

      1. fma-define 32.4%

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

      2. fma-define 32.4%

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

   3. Simplified 32.4%

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

      1. add-cube-cbrt 32.4%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

      2. add-sqr-sqrt 32.4%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. times-frac 32.4%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      4. pow2 32.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      5. fma-define 32.4%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      6. hypot-define 32.4%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

      7. fma-define 32.4%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}} \]

      8. hypot-define 55.9%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   7. Step-by-step derivation

      1. associate-*r/ 56.0%

         \[\leadsto \color{blue}{\frac{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

      2. clear-num 53.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}} \]

      3. associate-*l/ 53.7%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\color{blue}{\frac{{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}}} \]

      4. unpow2 53.7%

         \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}} \]

      5. add-cube-cbrt 54.0%

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

   8. Applied egg-rr 54.0%

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

   9. Taylor expanded in y.im around -inf 34.2%

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

       1. mul-1-neg 34.2%

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

   11. Simplified 34.2%

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

   12. Taylor expanded in y.re around -inf 31.6%

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

Alternative 12: 42.8% 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 63.3%

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

   1. fma-define 63.3%

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

   2. fma-define 63.3%

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

3. Simplified 63.3%

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

5. Taylor expanded in y.re around 0 41.3%

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

Reproduce

herbie shell --seed 2024145 
(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))))