_divideComplex, real part

Percentage Accurate: 62.5% → 82.9%

Time: 9.7s

Alternatives: 10

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: 62.5% 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: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.re + \frac{\mathsf{fma}\left(x.re, \frac{y.im \cdot y.im}{-y.re}, y.im \cdot x.im\right)}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.re}}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.im -9.8e+101)
     (/ (fma y.re (/ x.re y.im) x.im) y.im)
     (if (<= y.im -5e-76)
       t_0
       (if (<= y.im 5.2e-126)
         (/
          (+ x.re (/ (fma x.re (/ (* y.im y.im) (- y.re)) (* y.im x.im)) y.re))
          y.re)
         (if (<= y.im 3.3e+38)
           t_0
           (/ (fma y.re (/ 1.0 (/ y.im x.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(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.8e+101) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_im <= -5e-76) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-126) {
		tmp = (x_46_re + (fma(x_46_re, ((y_46_im * y_46_im) / -y_46_re), (y_46_im * x_46_im)) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.3e+38) {
		tmp = t_0;
	} else {
		tmp = fma(y_46_re, (1.0 / (y_46_im / x_46_re)), 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(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -9.8e+101)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_im <= -5e-76)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-126)
		tmp = Float64(Float64(x_46_re + Float64(fma(x_46_re, Float64(Float64(y_46_im * y_46_im) / Float64(-y_46_re)), Float64(y_46_im * x_46_im)) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.3e+38)
		tmp = t_0;
	else
		tmp = Float64(fma(y_46_re, Float64(1.0 / Float64(y_46_im / x_46_re)), 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[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+101], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -5e-76], t$95$0, If[LessEqual[y$46$im, 5.2e-126], N[(N[(x$46$re + N[(N[(x$46$re * N[(N[(y$46$im * y$46$im), $MachinePrecision] / (-y$46$re)), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.3e+38], t$95$0, N[(N[(y$46$re * N[(1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.79999999999999965e101

   1. Initial program 41.5%

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

      1. 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

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

      3. lower-fma.f64 41.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -9.79999999999999965e101 < y.im < -4.9999999999999998e-76 or 5.19999999999999998e-126 < y.im < 3.2999999999999999e38

   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. 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

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

      3. lower-fma.f64 82.4

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if -4.9999999999999998e-76 < y.im < 5.19999999999999998e-126

   1. Initial program 71.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 68.2

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \left(-1 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \frac{x.im \cdot y.im}{y.re}\right)}{y.re}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.re + \color{blue}{\left(\frac{x.im \cdot y.im}{y.re} + -1 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}}\right)}}{y.re} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \frac{x.re + \frac{x.im \cdot y.im + \color{blue}{-1 \cdot \frac{x.re \cdot {y.im}^{2}}{y.re}}}{y.re}}{y.re} \]

      9. +-commutative N/A

         \[\leadsto \frac{x.re + \frac{\color{blue}{-1 \cdot \frac{x.re \cdot {y.im}^{2}}{y.re} + x.im \cdot y.im}}{y.re}}{y.re} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x.re + \frac{-1 \cdot \frac{x.re \cdot {y.im}^{2}}{y.re} + x.im \cdot y.im}{y.re}}{y.re}} \]

   10. Applied rewrites 94.1%

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

   if 3.2999999999999999e38 < y.im

   1. Initial program 54.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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

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

      3. lower-fma.f64 54.8

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.4

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.re + \frac{\mathsf{fma}\left(x.re, \frac{y.im \cdot y.im}{-y.re}, y.im \cdot x.im\right)}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.re}}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.re}}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.im -9.8e+101)
     (/ (fma y.re (/ x.re y.im) x.im) y.im)
     (if (<= y.im -7.4e-76)
       t_0
       (if (<= y.im 1.1e-125)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 3.3e+38)
           t_0
           (/ (fma y.re (/ 1.0 (/ y.im x.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(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.8e+101) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_im <= -7.4e-76) {
		tmp = t_0;
	} else if (y_46_im <= 1.1e-125) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 3.3e+38) {
		tmp = t_0;
	} else {
		tmp = fma(y_46_re, (1.0 / (y_46_im / x_46_re)), 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(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -9.8e+101)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_im <= -7.4e-76)
		tmp = t_0;
	elseif (y_46_im <= 1.1e-125)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 3.3e+38)
		tmp = t_0;
	else
		tmp = Float64(fma(y_46_re, Float64(1.0 / Float64(y_46_im / x_46_re)), 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[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+101], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.4e-76], t$95$0, If[LessEqual[y$46$im, 1.1e-125], 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, 3.3e+38], t$95$0, N[(N[(y$46$re * N[(1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.79999999999999965e101

   1. Initial program 41.5%

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

      1. 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

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

      3. lower-fma.f64 41.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -9.79999999999999965e101 < y.im < -7.40000000000000023e-76 or 1.09999999999999997e-125 < y.im < 3.2999999999999999e38

   1. Initial program 82.1%

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

      1. 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

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

      3. lower-fma.f64 82.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 82.1

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

   4. Applied rewrites 82.1%

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

   if -7.40000000000000023e-76 < y.im < 1.09999999999999997e-125

   1. Initial program 71.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 3.2999999999999999e38 < y.im

   1. Initial program 54.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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

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

      3. lower-fma.f64 54.8

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.4

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{1}{\frac{y.im}{x.re}}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re y.re (* y.im x.im)) (fma y.re y.re (* y.im y.im)))))
   (if (<= y.im -9.8e+101)
     (/ (fma y.re (/ x.re y.im) x.im) y.im)
     (if (<= y.im -7.4e-76)
       t_0
       (if (<= y.im 1.1e-125)
         (/ (fma x.im (/ y.im y.re) x.re) y.re)
         (if (<= y.im 3.3e+38) t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.8e+101) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_im <= -7.4e-76) {
		tmp = t_0;
	} else if (y_46_im <= 1.1e-125) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 3.3e+38) {
		tmp = t_0;
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -9.8e+101)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_im <= -7.4e-76)
		tmp = t_0;
	elseif (y_46_im <= 1.1e-125)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 3.3e+38)
		tmp = t_0;
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.8e+101], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.4e-76], t$95$0, If[LessEqual[y$46$im, 1.1e-125], 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, 3.3e+38], t$95$0, N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.79999999999999965e101

   1. Initial program 41.5%

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

      1. 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

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

      3. lower-fma.f64 41.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if -9.79999999999999965e101 < y.im < -7.40000000000000023e-76 or 1.09999999999999997e-125 < y.im < 3.2999999999999999e38

   1. Initial program 82.1%

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

      1. 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

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

      3. lower-fma.f64 82.1

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 82.1

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

   4. Applied rewrites 82.1%

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

   if -7.40000000000000023e-76 < y.im < 1.09999999999999997e-125

   1. Initial program 71.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 3.2999999999999999e38 < y.im

   1. Initial program 54.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-44}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.9e+119)
   (/ x.im y.im)
   (if (<= y.im -9e-44)
     (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))
     (if (<= y.im 3.1e-116)
       (/ x.re y.re)
       (if (<= y.im 6.2e+106)
         (/ (fma x.re y.re (* y.im x.im)) (* y.im y.im))
         (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.9e+119) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9e-44) {
		tmp = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_im <= 3.1e-116) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 6.2e+106) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / (y_46_im * y_46_im);
	} 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 <= -4.9e+119)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -9e-44)
		tmp = Float64(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_im <= 3.1e-116)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 6.2e+106)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / Float64(y_46_im * y_46_im));
	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, -4.9e+119], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9e-44], 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], If[LessEqual[y$46$im, 3.1e-116], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.2e+106], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.89999999999999996e119 or 6.1999999999999999e106 < y.im

   1. Initial program 45.2%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -4.89999999999999996e119 < y.im < -8.9999999999999997e-44

   1. Initial program 80.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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

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

      3. lower-fma.f64 80.0

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 80.0

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

   4. Applied rewrites 80.0%

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

   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 77.0

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

   7. Applied rewrites 77.0%

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

   if -8.9999999999999997e-44 < y.im < 3.10000000000000018e-116

   1. Initial program 72.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   if 3.10000000000000018e-116 < y.im < 6.1999999999999999e106

   1. Initial program 80.2%

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

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

      3. lower-fma.f64 80.2

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   7. Applied rewrites 61.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-44}:\\ \;\;\;\;x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-116}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.im (/ y.im (fma y.im y.im (* y.re y.re))))))
   (if (<= y.im -4.9e+119)
     (/ x.im y.im)
     (if (<= y.im -9e-44)
       t_0
       (if (<= y.im 2.35e-104)
         (/ x.re y.re)
         (if (<= y.im 5.8e+144) 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 = x_46_im * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	double tmp;
	if (y_46_im <= -4.9e+119) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9e-44) {
		tmp = t_0;
	} else if (y_46_im <= 2.35e-104) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 5.8e+144) {
		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(x_46_im * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_im <= -4.9e+119)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -9e-44)
		tmp = t_0;
	elseif (y_46_im <= 2.35e-104)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 5.8e+144)
		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[(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]}, If[LessEqual[y$46$im, -4.9e+119], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9e-44], t$95$0, If[LessEqual[y$46$im, 2.35e-104], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.8e+144], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.89999999999999996e119 or 5.79999999999999996e144 < y.im

   1. Initial program 41.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -4.89999999999999996e119 < y.im < -8.9999999999999997e-44 or 2.35e-104 < y.im < 5.79999999999999996e144

   1. Initial program 77.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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

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

      3. lower-fma.f64 77.9

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 77.9

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

   4. Applied rewrites 77.9%

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

   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 67.6

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

   7. Applied rewrites 67.6%

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

   if -8.9999999999999997e-44 < y.im < 2.35e-104

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 6: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -350000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -350000000000.0)
   (/ x.im y.im)
   (if (<= y.im 5.7e-84)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (if (<= y.im 6.2e+106)
       (/ (fma x.re y.re (* y.im x.im)) (* y.im y.im))
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -350000000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.7e-84) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else if (y_46_im <= 6.2e+106) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / (y_46_im * y_46_im);
	} 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 <= -350000000000.0)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 5.7e-84)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	elseif (y_46_im <= 6.2e+106)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / Float64(y_46_im * y_46_im));
	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, -350000000000.0], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.7e-84], 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, 6.2e+106], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.5e11 or 6.1999999999999999e106 < y.im

   1. Initial program 52.1%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -3.5e11 < y.im < 5.7e-84

   1. Initial program 74.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. 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 87.0

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

   5. Applied rewrites 87.0%

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

   if 5.7e-84 < y.im < 6.1999999999999999e106

   1. Initial program 78.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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

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

      3. lower-fma.f64 78.8

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 78.8

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.3

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

   7. Applied rewrites 65.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -350000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -165000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -165000000000.0)
   (/ (fma y.re (/ x.re y.im) x.im) y.im)
   (if (<= y.im 5.7e-84)
     (/ (fma x.im (/ y.im y.re) x.re) y.re)
     (/ (fma x.re (/ y.re y.im) x.im) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -165000000000.0) {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im;
	} else if (y_46_im <= 5.7e-84) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), x_46_re) / y_46_re;
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -165000000000.0)
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im);
	elseif (y_46_im <= 5.7e-84)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), x_46_re) / y_46_re);
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -165000000000.0], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.7e-84], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.65e11

   1. Initial program 54.3%

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

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

      3. lower-fma.f64 54.3

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 54.3

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in y.im around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 81.6

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

   7. Applied rewrites 81.6%

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

   if -1.65e11 < y.im < 5.7e-84

   1. Initial program 74.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. 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 87.0

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

   5. Applied rewrites 87.0%

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

   if 5.7e-84 < 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

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 82.2

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

   5. Applied rewrites 82.2%

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

Alternative 8: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -165000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.im}{y.re}, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.re (/ y.re y.im) x.im) y.im)))
   (if (<= y.im -165000000000.0)
     t_0
     (if (<= y.im 5.7e-84) (/ (fma x.im (/ y.im y.re) 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_re / y_46_im), x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -165000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 5.7e-84) {
		tmp = fma(x_46_im, (y_46_im / y_46_re), 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(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -165000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 5.7e-84)
		tmp = Float64(fma(x_46_im, Float64(y_46_im / y_46_re), 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[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -165000000000.0], t$95$0, If[LessEqual[y$46$im, 5.7e-84], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.65e11 or 5.7e-84 < y.im

   1. Initial program 58.2%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 81.3

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

   5. Applied rewrites 81.3%

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

   if -1.65e11 < y.im < 5.7e-84

   1. Initial program 74.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. 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 87.0

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

   5. Applied rewrites 87.0%

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

Alternative 9: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3e-43)
   (/ x.im y.im)
   (if (<= y.im 2e-103) (/ 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-43) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-103) {
		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-43)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2d-103) 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-43) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2e-103) {
		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-43:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2e-103:
		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-43)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2e-103)
		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-43)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2e-103)
		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-43], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2e-103], 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 < -3.00000000000000003e-43 or 1.99999999999999992e-103 < 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.re around 0

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -3.00000000000000003e-43 < y.im < 1.99999999999999992e-103

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 10: 42.7% 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 65.8%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 45.5

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

5. Applied rewrites 45.5%

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

Reproduce

herbie shell --seed 2024219 
(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))))