_divideComplex, real part

Percentage Accurate: 61.5% → 82.2%

Time: 9.0s

Alternatives: 10

Speedup: 0.9×

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: 61.5% 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: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.im x.im (* x.re y.re)) (fma y.im y.im (* y.re y.re)))))
   (if (<= y.re -1.62e+97)
     (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
     (if (<= y.re -1.2e-46)
       t_0
       (if (<= y.re 8.5e-122)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.7e+96) t_0 (/ (fma (/ y.im y.re) x.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 t_0 = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -1.62e+97) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_re <= -1.2e-46) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e-122) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.62e+97)
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.2e-46)
		tmp = t_0;
	elseif (y_46_re <= 8.5e-122)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.7e+96)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.62e+97], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.2e-46], t$95$0, If[LessEqual[y$46$re, 8.5e-122], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+96], t$95$0, N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.62e97

   1. Initial program 35.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 88.2%

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

   if -1.62e97 < y.re < -1.20000000000000007e-46 or 8.50000000000000003e-122 < y.re < 1.7e96

   1. Initial program 74.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if -1.20000000000000007e-46 < y.re < 8.50000000000000003e-122

   1. Initial program 68.4%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 68.4

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

   4. Applied rewrites 68.4%

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

   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if 1.7e96 < y.re

   1. Initial program 28.1%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 28.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 28.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 28.1

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   6. 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. *-commutative 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 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.im x.im (* x.re y.re)) (fma y.im y.im (* y.re y.re)))))
   (if (<= y.re -1.62e+97)
     (/ (fma (/ x.im y.re) y.im x.re) y.re)
     (if (<= y.re -1.2e-46)
       t_0
       (if (<= y.re 8.5e-122)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.7e+96) t_0 (/ (fma (/ y.im y.re) x.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 t_0 = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -1.62e+97) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -1.2e-46) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e-122) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.62e+97)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -1.2e-46)
		tmp = t_0;
	elseif (y_46_re <= 8.5e-122)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.7e+96)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.62e+97], 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, -1.2e-46], t$95$0, If[LessEqual[y$46$re, 8.5e-122], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+96], t$95$0, N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.62e97

   1. Initial program 35.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 88.2

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

   5. Applied rewrites 88.2%

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

   if -1.62e97 < y.re < -1.20000000000000007e-46 or 8.50000000000000003e-122 < y.re < 1.7e96

   1. Initial program 74.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 74.7

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

   4. Applied rewrites 74.7%

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

   if -1.20000000000000007e-46 < y.re < 8.50000000000000003e-122

   1. Initial program 68.4%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 68.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 68.4

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

   4. Applied rewrites 68.4%

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

   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if 1.7e96 < y.re

   1. Initial program 28.1%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 28.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 28.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 28.1

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

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   6. 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. *-commutative 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 84.9

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

   7. Applied rewrites 84.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.62 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.35e+112)
   (/ x.im y.im)
   (if (<= y.im -2.8e-65)
     (* (/ y.im (fma y.im y.im (* y.re y.re))) x.im)
     (if (<= y.im 7.8e+141)
       (/ (fma (/ x.im y.re) y.im 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 <= -3.35e+112) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -2.8e-65) {
		tmp = (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else if (y_46_im <= 7.8e+141) {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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 <= -3.35e+112)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -2.8e-65)
		tmp = Float64(Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	elseif (y_46_im <= 7.8e+141)
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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_] := If[LessEqual[y$46$im, -3.35e+112], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -2.8e-65], N[(N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.8e+141], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.3499999999999999e112 or 7.79999999999999983e141 < y.im

   1. Initial program 28.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if -3.3499999999999999e112 < y.im < -2.8e-65

   1. Initial program 81.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if -2.8e-65 < y.im < 7.79999999999999983e141

   1. Initial program 67.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. *-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 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 4: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-253}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.35e+112)
   (/ x.im y.im)
   (if (<= y.im -5.6e-71)
     (* (/ y.im (fma y.im y.im (* y.re y.re))) x.im)
     (if (<= y.im -3.4e-253)
       (/ (fma y.im x.im (* x.re y.re)) (* y.re y.re))
       (if (<= y.im 5.2e+141) (/ 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 <= -3.35e+112) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -5.6e-71) {
		tmp = (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else if (y_46_im <= -3.4e-253) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / (y_46_re * y_46_re);
	} else if (y_46_im <= 5.2e+141) {
		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 <= -3.35e+112)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -5.6e-71)
		tmp = Float64(Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	elseif (y_46_im <= -3.4e-253)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 5.2e+141)
		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_] := If[LessEqual[y$46$im, -3.35e+112], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -5.6e-71], N[(N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.4e-253], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+141], 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 < -3.3499999999999999e112 or 5.1999999999999999e141 < y.im

   1. Initial program 28.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if -3.3499999999999999e112 < y.im < -5.60000000000000001e-71

   1. Initial program 79.5%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.5

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 79.5

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.1

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

   7. Applied rewrites 67.1%

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

   if -5.60000000000000001e-71 < y.im < -3.39999999999999985e-253

   1. Initial program 82.9%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 82.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 82.9

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in y.im around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   if -3.39999999999999985e-253 < y.im < 5.1999999999999999e141

   1. Initial program 62.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 62.8

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

   5. Applied rewrites 62.8%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-253}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -250000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -250000.0)
   (/ (fma (/ y.im y.re) x.im x.re) y.re)
   (if (<= y.re 6.5e+50)
     (/ (fma (/ y.re y.im) x.re x.im) y.im)
     (/ (fma (/ x.im y.re) 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 <= -250000.0) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= 6.5e+50) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((x_46_im / y_46_re), y_46_im, 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 <= -250000.0)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= 6.5e+50)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, 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, -250000.0], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.5e+50], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.5e5

   1. Initial program 48.3%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 48.3

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 48.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 48.3

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

   4. Applied rewrites 48.3%

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

   5. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   6. 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. *-commutative 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 81.5

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

   7. Applied rewrites 81.5%

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

   if -2.5e5 < y.re < 6.5000000000000003e50

   1. Initial program 70.1%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 70.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 70.1

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

   4. Applied rewrites 70.1%

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

   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 77.8

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

   7. Applied rewrites 77.8%

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

   if 6.5000000000000003e50 < y.re

   1. Initial program 36.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 6: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.7e-7)
     t_0
     (if (<= y.im 5.2e+141) (/ (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 <= -1.7e-7) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e+141) {
		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 <= -1.7e-7)
		tmp = t_0;
	elseif (y_46_im <= 5.2e+141)
		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, -1.7e-7], t$95$0, If[LessEqual[y$46$im, 5.2e+141], 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 < -1.69999999999999987e-7 or 5.1999999999999999e141 < y.im

   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.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 80.6

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

   5. Applied rewrites 80.6%

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

   if -1.69999999999999987e-7 < y.im < 5.1999999999999999e141

   1. Initial program 69.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 78.2

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

   5. Applied rewrites 78.2%

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

Alternative 7: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.35 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.35e+112)
   (/ x.im y.im)
   (if (<= y.im -8.2e-67)
     (* (/ y.im (fma y.im y.im (* y.re y.re))) x.im)
     (if (<= y.im 5.2e+141) (/ 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 <= -3.35e+112) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -8.2e-67) {
		tmp = (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else if (y_46_im <= 5.2e+141) {
		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 <= -3.35e+112)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -8.2e-67)
		tmp = Float64(Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	elseif (y_46_im <= 5.2e+141)
		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_] := If[LessEqual[y$46$im, -3.35e+112], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-67], N[(N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+141], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.3499999999999999e112 or 5.1999999999999999e141 < y.im

   1. Initial program 28.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if -3.3499999999999999e112 < y.im < -8.1999999999999994e-67

   1. Initial program 81.7%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 81.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.8

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

   7. Applied rewrites 68.8%

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

   if -8.1999999999999994e-67 < y.im < 5.1999999999999999e141

   1. Initial program 67.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 60.3

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

   5. Applied rewrites 60.3%

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

Alternative 8: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.im\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.8e+115)
   (/ x.im y.im)
   (if (<= y.im -8.5e-67)
     (* (/ x.im (fma y.im y.im (* y.re y.re))) y.im)
     (if (<= y.im 5.2e+141) (/ 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 <= -2.8e+115) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -8.5e-67) {
		tmp = (x_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * y_46_im;
	} else if (y_46_im <= 5.2e+141) {
		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 <= -2.8e+115)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -8.5e-67)
		tmp = Float64(Float64(x_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * y_46_im);
	elseif (y_46_im <= 5.2e+141)
		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_] := If[LessEqual[y$46$im, -2.8e+115], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -8.5e-67], N[(N[(x$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+141], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.8e115 or 5.1999999999999999e141 < y.im

   1. Initial program 27.1%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.8e115 < y.im < -8.49999999999999993e-67

   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. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if -8.49999999999999993e-67 < y.im < 5.1999999999999999e141

   1. Initial program 67.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 60.3

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

   5. Applied rewrites 60.3%

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

Alternative 9: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.6e-65)
   (/ x.im y.im)
   (if (<= y.im 5.2e+141) (/ 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 <= -2.6e-65) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.2e+141) {
		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 <= (-2.6d-65)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 5.2d+141) 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 <= -2.6e-65) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 5.2e+141) {
		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 <= -2.6e-65:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 5.2e+141:
		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 <= -2.6e-65)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 5.2e+141)
		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 <= -2.6e-65)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 5.2e+141)
		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, -2.6e-65], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+141], 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 < -2.6000000000000001e-65 or 5.1999999999999999e141 < y.im

   1. Initial program 45.6%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-/.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -2.6000000000000001e-65 < y.im < 5.1999999999999999e141

   1. Initial program 67.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 60.3

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

   5. Applied rewrites 60.3%

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

Alternative 10: 43.5% 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 57.9%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 41.8

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

5. Applied rewrites 41.8%

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

Reproduce

herbie shell --seed 2024243 
(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))))