_divideComplex, real part

Percentage Accurate: 61.7% → 77.3%

Time: 8.5s

Alternatives: 9

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: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\frac{t\_0}{y.re}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ x.im y.re) y.im x.re)))
   (if (<= y.re -1.45e+65)
     (/ t_0 y.re)
     (if (<= y.re 2.3e+100)
       (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.im y.im) (* y.re y.re)))
       (/ 1.0 (/ 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_im / y_46_re), y_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -1.45e+65) {
		tmp = t_0 / y_46_re;
	} else if (y_46_re <= 2.3e+100) {
		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 = 1.0 / (y_46_re / t_0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re)
	tmp = 0.0
	if (y_46_re <= -1.45e+65)
		tmp = Float64(t_0 / y_46_re);
	elseif (y_46_re <= 2.3e+100)
		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 = Float64(1.0 / Float64(y_46_re / 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 / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+65], N[(t$95$0 / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+100], 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], N[(1.0 / N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.45e65

   1. Initial program 36.9%

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

   3. Taylor expanded in y.re around 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 92.0

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

   5. Applied rewrites 92.0%

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

   if -1.45e65 < y.re < 2.2999999999999999e100

   1. Initial program 82.4%

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

   if 2.2999999999999999e100 < y.re

   1. Initial program 46.4%

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

   3. Taylor expanded in y.re around 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 85.4%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re}{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.im -4.5e+15)
     (/ x.im y.im)
     (if (<= y.im 1.05e-38)
       t_0
       (if (<= y.im 6.8e+58)
         (/ (fma y.re x.re (* y.im x.im)) (* y.im y.im))
         (if (<= y.im 4.2e+89) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_im <= -4.5e+15) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.05e-38) {
		tmp = t_0;
	} else if (y_46_im <= 6.8e+58) {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 4.2e+89) {
		tmp = t_0;
	} 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)
	t_0 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_im <= -4.5e+15)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.05e-38)
		tmp = t_0;
	elseif (y_46_im <= 6.8e+58)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 4.2e+89)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$im, -4.5e+15], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.05e-38], t$95$0, If[LessEqual[y$46$im, 6.8e+58], N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e+89], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.5e15 or 4.19999999999999972e89 < y.im

   1. Initial program 60.9%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -4.5e15 < y.im < 1.05000000000000006e-38 or 6.8000000000000001e58 < y.im < 4.19999999999999972e89

   1. Initial program 70.9%

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

   3. Taylor expanded in y.re around 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 1.05000000000000006e-38 < y.im < 6.8000000000000001e58

   1. Initial program 95.5%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{y.im}^{2}}} \]
   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 78.9

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

   5. Applied rewrites 78.9%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 78.9

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

   7. Applied rewrites 78.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -1.45e+65)
     t_0
     (if (<= y.re 2.3e+100)
       (/ (+ (* 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 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -1.45e+65) {
		tmp = t_0;
	} else if (y_46_re <= 2.3e+100) {
		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(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.45e+65)
		tmp = t_0;
	elseif (y_46_re <= 2.3e+100)
		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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+65], t$95$0, If[LessEqual[y$46$re, 2.3e+100], 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 2 regimes

2. if y.re < -1.45e65 or 2.2999999999999999e100 < y.re

   1. Initial program 40.8%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -1.45e65 < y.re < 2.2999999999999999e100

   1. Initial program 82.4%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re x.re (* y.im x.im))))
   (if (<= y.im -4.4e-66)
     (/ x.im y.im)
     (if (<= y.im 3.8e-87)
       (/ t_0 (* y.re y.re))
       (if (<= y.im 8.8e+58)
         (/ t_0 (* y.im y.im))
         (if (<= y.im 3.8e+89) (/ 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 t_0 = fma(y_46_re, x_46_re, (y_46_im * x_46_im));
	double tmp;
	if (y_46_im <= -4.4e-66) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.8e-87) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else if (y_46_im <= 8.8e+58) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_im <= 3.8e+89) {
		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)
	t_0 = fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_im <= -4.4e-66)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 3.8e-87)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 8.8e+58)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 3.8e+89)
		tmp = Float64(x_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_] := Block[{t$95$0 = N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.4e-66], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.8e-87], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.8e+58], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+89], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.4000000000000002e-66 or 3.80000000000000023e89 < y.im

   1. Initial program 60.5%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.4000000000000002e-66 < y.im < 3.8e-87

   1. Initial program 75.8%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{y.im}^{2}}} \]
   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 24.7

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

   5. Applied rewrites 24.7%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 24.7

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in y.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

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

   if 3.8e-87 < y.im < 8.8000000000000003e58

   1. Initial program 88.2%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{y.im}^{2}}} \]
   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 68.9

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

   5. Applied rewrites 68.9%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 68.9

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

   7. Applied rewrites 68.9%

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

   if 8.8000000000000003e58 < y.im < 3.80000000000000023e89

   1. Initial program 18.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 85.0

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

   5. Applied rewrites 85.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.04 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -4.2e+15)
     t_0
     (if (<= y.im 1.04e-38) (/ (fma (/ y.im y.re) x.im x.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_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -4.2e+15) {
		tmp = t_0;
	} else if (y_46_im <= 1.04e-38) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_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(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.2e+15)
		tmp = t_0;
	elseif (y_46_im <= 1.04e-38)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_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[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.2e+15], t$95$0, If[LessEqual[y$46$im, 1.04e-38], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.2e15 or 1.04e-38 < y.im

   1. Initial program 65.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -4.2e15 < y.im < 1.04e-38

   1. Initial program 73.4%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 18.9

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

   5. Applied rewrites 18.9%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   7. 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. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.8

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

   8. Applied rewrites 82.8%

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

Alternative 6: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.04 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.re y.im) y.re x.im) y.im)))
   (if (<= y.im -4.2e+15)
     t_0
     (if (<= y.im 1.04e-38) (/ (fma (/ x.im y.re) y.im x.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_im), y_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -4.2e+15) {
		tmp = t_0;
	} else if (y_46_im <= 1.04e-38) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_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(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.2e+15)
		tmp = t_0;
	elseif (y_46_im <= 1.04e-38)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_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[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.2e+15], t$95$0, If[LessEqual[y$46$im, 1.04e-38], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.2e15 or 1.04e-38 < y.im

   1. Initial program 65.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -4.2e15 < y.im < 1.04e-38

   1. Initial program 73.4%

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

   3. Taylor expanded in y.re around 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

Alternative 7: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.05e-7)
   (/ x.re y.re)
   (if (<= y.re 2.05e+54)
     (/ (fma y.re x.re (* y.im x.im)) (* y.im y.im))
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.05e-7) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.05e+54) {
		tmp = fma(y_46_re, x_46_re, (y_46_im * x_46_im)) / (y_46_im * y_46_im);
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.05e-7)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.05e+54)
		tmp = Float64(fma(y_46_re, x_46_re, Float64(y_46_im * x_46_im)) / Float64(y_46_im * y_46_im));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.05e-7], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.05e+54], N[(N[(y$46$re * x$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.05e-7 or 2.04999999999999984e54 < y.re

   1. Initial program 48.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -1.05e-7 < y.re < 2.04999999999999984e54

   1. Initial program 81.9%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{{y.im}^{2}}} \]
   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 63.2

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

   5. Applied rewrites 63.2%

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 63.3

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

   7. Applied rewrites 63.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.2e-9)
   (/ x.re y.re)
   (if (<= y.re 2.05e+54) (/ 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_re <= -3.2e-9) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.05e+54) {
		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_46re <= (-3.2d-9)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.05d+54) 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_re <= -3.2e-9) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.05e+54) {
		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_re <= -3.2e-9:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.05e+54:
		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_re <= -3.2e-9)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.05e+54)
		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_re <= -3.2e-9)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.05e+54)
		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[LessEqual[y$46$re, -3.2e-9], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.05e+54], 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.re < -3.20000000000000012e-9 or 2.04999999999999984e54 < y.re

   1. Initial program 49.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -3.20000000000000012e-9 < y.re < 2.04999999999999984e54

   1. Initial program 81.8%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

Alternative 9: 42.3% 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 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.im around inf

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

   1. lower-/.f64 44.6

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

5. Applied rewrites 44.6%

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

Reproduce

herbie shell --seed 2024264 
(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))))