_divideComplex, real part

Percentage Accurate: 61.4% → 80.4%

Time: 10.7s

Alternatives: 8

Speedup: 1.6×

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: 61.4% 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: 80.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -7e-29)
     t_0
     (if (<= y.im 4.25e-106)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (if (<= y.im 2.9e+60)
         (/ (+ (* x.re y.re) (* y.im x.im)) (+ (* 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 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-29) {
		tmp = t_0;
	} else if (y_46_im <= 4.25e-106) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 2.9e+60) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((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(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-29)
		tmp = t_0;
	elseif (y_46_im <= 4.25e-106)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 2.9e+60)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(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$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-29], t$95$0, If[LessEqual[y$46$im, 4.25e-106], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.9e+60], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.9999999999999995e-29 or 2.9e60 < y.im

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -6.9999999999999995e-29 < y.im < 4.2499999999999999e-106

   1. Initial program 69.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 4.2499999999999999e-106 < y.im < 2.9e60

   1. Initial program 94.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 4.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.3e+82)
   (/ x.im y.im)
   (if (<= y.im -7e-29)
     (/ (fma y.im x.im (* x.re y.re)) (* y.im y.im))
     (if (<= y.im 2.1e-36)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (if (<= y.im 7.7e+130)
         (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
         (/ 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 <= -2.3e+82) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -7e-29) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.1e-36) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 7.7e+130) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = 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 <= -2.3e+82)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -7e-29)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.1e-36)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 7.7e+130)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(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, -2.3e+82], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7e-29], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.1e-36], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.7e+130], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.29999999999999988e82 or 7.7000000000000004e130 < y.im

   1. Initial program 30.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -2.29999999999999988e82 < y.im < -6.9999999999999995e-29

   1. Initial program 78.3%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 69.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.8

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

   7. Applied rewrites 69.8%

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

   if -6.9999999999999995e-29 < y.im < 2.09999999999999991e-36

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 2.09999999999999991e-36 < y.im < 7.7000000000000004e130

   1. Initial program 73.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 73.4

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 73.4

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 73.4

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

   4. Applied rewrites 73.4%

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

   5. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.5

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

   7. Applied rewrites 72.5%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.3e+82)
   (/ x.im y.im)
   (if (<= y.im -3.1e-29)
     (/ (fma y.im x.im (* x.re y.re)) (* y.im y.im))
     (if (<= y.im 2.05e-120)
       (/ x.re y.re)
       (if (<= y.im 7.7e+130)
         (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
         (/ 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 <= -2.3e+82) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -3.1e-29) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.05e-120) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 7.7e+130) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = 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 <= -2.3e+82)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -3.1e-29)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.05e-120)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 7.7e+130)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(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, -2.3e+82], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.1e-29], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.05e-120], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.7e+130], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.29999999999999988e82 or 7.7000000000000004e130 < y.im

   1. Initial program 30.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -2.29999999999999988e82 < y.im < -3.10000000000000026e-29

   1. Initial program 78.3%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 69.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 69.8

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

   7. Applied rewrites 69.8%

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

   if -3.10000000000000026e-29 < y.im < 2.05000000000000017e-120

   1. Initial program 71.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if 2.05000000000000017e-120 < y.im < 7.7000000000000004e130

   1. Initial program 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 75.3

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 75.3

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -7e-29)
     t_0
     (if (<= y.im 2.1e-36)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (if (<= y.im 7.5e+79) (/ x.im (+ y.im (/ (* y.re 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 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-29) {
		tmp = t_0;
	} else if (y_46_im <= 2.1e-36) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 7.5e+79) {
		tmp = x_46_im / (y_46_im + ((y_46_re * 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(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-29)
		tmp = t_0;
	elseif (y_46_im <= 2.1e-36)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 7.5e+79)
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(Float64(y_46_re * 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$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-29], t$95$0, If[LessEqual[y$46$im, 2.1e-36], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+79], N[(x$46$im / N[(y$46$im + N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.9999999999999995e-29 or 7.49999999999999967e79 < y.im

   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. Add Preprocessing

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -6.9999999999999995e-29 < y.im < 2.09999999999999991e-36

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 2.09999999999999991e-36 < y.im < 7.49999999999999967e79

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 86.2

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 86.2

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 86.2

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

   4. Applied rewrites 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-in N/A

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

      12. lift-/.f64 N/A

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

      13. div-inv N/A

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

      14. lift-*.f64 N/A

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

      15. *-rgt-identity N/A

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

      16. times-frac N/A

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

      17. /-rgt-identity N/A

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

   6. Applied rewrites 86.2%

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

   7. Taylor expanded in x.im around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.0

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

   9. Applied rewrites 84.0%

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

Alternative 5: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -7e-29)
     t_0
     (if (<= y.im 2.1e-36)
       (/ (fma x.im (/ y.im y.re) x.re) y.re)
       (if (<= y.im 7.5e+79)
         (* x.im (/ y.im (fma 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 = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-29) {
		tmp = t_0;
	} else if (y_46_im <= 2.1e-36) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 7.5e+79) {
		tmp = x_46_im * (y_46_im / fma(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(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-29)
		tmp = t_0;
	elseif (y_46_im <= 2.1e-36)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 7.5e+79)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-29], t$95$0, If[LessEqual[y$46$im, 2.1e-36], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+79], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.9999999999999995e-29 or 7.49999999999999967e79 < y.im

   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. Add Preprocessing

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -6.9999999999999995e-29 < y.im < 2.09999999999999991e-36

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 2.09999999999999991e-36 < y.im < 7.49999999999999967e79

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 86.2

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 86.2

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

Alternative 6: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3e-20)
   (/ x.im y.im)
   (if (<= y.im 2.05e-120)
     (/ x.re y.re)
     (if (<= y.im 7.7e+130)
       (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
       (/ 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 <= -3e-20) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.05e-120) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 7.7e+130) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = 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 <= -3e-20)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.05e-120)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 7.7e+130)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(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, -3e-20], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.05e-120], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.7e+130], N[(x$46$im * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.00000000000000029e-20 or 7.7000000000000004e130 < y.im

   1. Initial program 40.4%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -3.00000000000000029e-20 < y.im < 2.05000000000000017e-120

   1. Initial program 71.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if 2.05000000000000017e-120 < y.im < 7.7000000000000004e130

   1. Initial program 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 75.3

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 75.3

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in x.re around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.2

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

   7. Applied rewrites 67.2%

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

Alternative 7: 62.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3e-20)
   (/ x.im y.im)
   (if (<= y.im 5.1e-120) (/ x.re y.re) (/ 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 <= -3e-20) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.1e-120) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_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-20)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 5.1d-120) then
        tmp = x_46re / y_46re
    else
        tmp = x_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-20) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.1e-120) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_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-20:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 5.1e-120:
		tmp = x_46_re / y_46_re
	else:
		tmp = 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 <= -3e-20)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 5.1e-120)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_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-20)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 5.1e-120)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_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-20], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.1e-120], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.00000000000000029e-20 or 5.0999999999999998e-120 < y.im

   1. Initial program 53.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -3.00000000000000029e-20 < y.im < 5.0999999999999998e-120

   1. Initial program 71.8%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.9

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

   5. Applied rewrites 71.9%

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

Alternative 8: 42.9% accurate, 3.2× 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 59.9%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 45.4

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

5. Applied rewrites 45.4%

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

Reproduce

herbie shell --seed 2024216 
(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))))