_divideComplex, real part

Percentage Accurate: 62.0% → 82.6%

Time: 7.7s

Alternatives: 8

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: 62.0% 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: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-146}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.im x.im (* y.re x.re)) (fma y.im y.im (* y.re y.re)))))
   (if (<= y.re -2.4e+28)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (if (<= y.re -6e-146)
       t_0
       (if (<= y.re 1.35e-103)
         (/ (fma (/ y.re y.im) x.re x.im) y.im)
         (if (<= y.re 1.2e+114)
           t_0
           (/ (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 t_0 = fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_re <= -2.4e+28) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_re <= -6e-146) {
		tmp = t_0;
	} else if (y_46_re <= 1.35e-103) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else if (y_46_re <= 1.2e+114) {
		tmp = t_0;
	} 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)
	t_0 = Float64(fma(y_46_im, x_46_im, Float64(y_46_re * x_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.4e+28)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_re <= -6e-146)
		tmp = t_0;
	elseif (y_46_re <= 1.35e-103)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	elseif (y_46_re <= 1.2e+114)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y$46$im * x$46$im + N[(y$46$re * x$46$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, -2.4e+28], 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, -6e-146], t$95$0, If[LessEqual[y$46$re, 1.35e-103], 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, 1.2e+114], t$95$0, 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 4 regimes

2. if y.re < -2.39999999999999981e28

   1. Initial program 40.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. +-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 40.2

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

      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 40.2

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

      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 40.2

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

   4. Applied rewrites 40.2%

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

   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 82.5

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

   7. Applied rewrites 82.5%

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

   if -2.39999999999999981e28 < y.re < -6.00000000000000038e-146 or 1.35000000000000005e-103 < y.re < 1.2e114

   1. Initial program 86.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 86.4

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

      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 86.4

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

      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 86.4

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

   4. Applied rewrites 86.4%

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

   if -6.00000000000000038e-146 < y.re < 1.35000000000000005e-103

   1. Initial program 65.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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 65.7

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

      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 65.7

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

      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 65.7

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in 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 93.5

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

   7. Applied rewrites 93.5%

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

   if 1.2e114 < y.re

   1. Initial program 34.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-174}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+127}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))))
   (if (<= y.re -1.45e+37)
     (/ x.re y.re)
     (if (<= y.re -1.4e-165)
       t_0
       (if (<= y.re 1.05e-174)
         (/ x.im y.im)
         (if (<= y.re 6.4e+127) t_0 (/ 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 = 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 <= -1.45e+37) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.4e-165) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-174) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 6.4e+127) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.45e+37)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.4e-165)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-174)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 6.4e+127)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+37], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.4e-165], t$95$0, If[LessEqual[y$46$re, 1.05e-174], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.4e+127], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.44999999999999989e37 or 6.39999999999999952e127 < y.re

   1. Initial program 37.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.44999999999999989e37 < y.re < -1.4e-165 or 1.05000000000000005e-174 < y.re < 6.39999999999999952e127

   1. Initial program 80.8%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 66.3%

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

   if -1.4e-165 < y.re < 1.05000000000000005e-174

   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.re around 0

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

      1. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-174}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+127}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.re) y.im x.re) y.re)))
   (if (<= y.re -6.8e+39)
     t_0
     (if (<= y.re -1.4e-165)
       (* x.re (/ y.re (fma y.im y.im (* y.re y.re))))
       (if (<= y.re 1.36e-62) (/ x.im y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	double tmp;
	if (y_46_re <= -6.8e+39) {
		tmp = t_0;
	} else if (y_46_re <= -1.4e-165) {
		tmp = x_46_re * (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else if (y_46_re <= 1.36e-62) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -6.8e+39)
		tmp = t_0;
	elseif (y_46_re <= -1.4e-165)
		tmp = Float64(x_46_re * Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 1.36e-62)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -6.8e+39], t$95$0, If[LessEqual[y$46$re, -1.4e-165], N[(x$46$re * N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.36e-62], N[(x$46$im / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.7999999999999998e39 or 1.35999999999999999e-62 < y.re

   1. Initial program 49.6%

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

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

   5. Applied rewrites 81.3%

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

   if -6.7999999999999998e39 < y.re < -1.4e-165

   1. Initial program 74.1%

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

   3. Taylor expanded in x.re around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   7. Applied rewrites 69.2%

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

   if -1.4e-165 < y.re < 1.35999999999999999e-62

   1. Initial program 67.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-165}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 1.36 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-146} \lor \neg \left(y.re \leq 2.8 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e-146) (not (<= y.re 2.8e-76)))
   (/ (fma (/ y.im y.re) x.im x.re) y.re)
   (/ (fma (/ y.re 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 tmp;
	if ((y_46_re <= -6e-146) || !(y_46_re <= 2.8e-76)) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((y_46_re / 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)
	tmp = 0.0
	if ((y_46_re <= -6e-146) || !(y_46_re <= 2.8e-76))
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / 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_] := If[Or[LessEqual[y$46$re, -6e-146], N[Not[LessEqual[y$46$re, 2.8e-76]], $MachinePrecision]], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.00000000000000038e-146 or 2.8000000000000001e-76 < y.re

   1. Initial program 55.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 55.9

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

      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 55.9

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

      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 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in y.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 76.3

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

   7. Applied rewrites 76.3%

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

   if -6.00000000000000038e-146 < y.re < 2.8000000000000001e-76

   1. Initial program 66.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 66.5

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

      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 66.5

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

      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 66.5

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

   4. Applied rewrites 66.5%

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

   5. Taylor expanded in 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 91.8

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

   7. Applied rewrites 91.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-146} \lor \neg \left(y.re \leq 2.8 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-146} \lor \neg \left(y.re \leq 2.8 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6e-146) (not (<= y.re 2.8e-76)))
   (/ (fma (/ y.im y.re) x.im x.re) y.re)
   (/ (fma (/ x.re y.im) 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_re <= -6e-146) || !(y_46_re <= 2.8e-76)) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((x_46_re / y_46_im), y_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)
	tmp = 0.0
	if ((y_46_re <= -6e-146) || !(y_46_re <= 2.8e-76))
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_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_] := If[Or[LessEqual[y$46$re, -6e-146], N[Not[LessEqual[y$46$re, 2.8e-76]], $MachinePrecision]], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.00000000000000038e-146 or 2.8000000000000001e-76 < y.re

   1. Initial program 55.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 55.9

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

      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 55.9

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

      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 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in y.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 76.3

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

   7. Applied rewrites 76.3%

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

   if -6.00000000000000038e-146 < y.re < 2.8000000000000001e-76

   1. Initial program 66.5%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}} \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{-146} \lor \neg \left(y.re \leq 2.8 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+30} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.3e+30) (not (<= y.re 3.5e-62)))
   (/ (fma (/ x.im y.re) y.im x.re) y.re)
   (/ (fma (/ x.re y.im) 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_re <= -2.3e+30) || !(y_46_re <= 3.5e-62)) {
		tmp = fma((x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re;
	} else {
		tmp = fma((x_46_re / y_46_im), y_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)
	tmp = 0.0
	if ((y_46_re <= -2.3e+30) || !(y_46_re <= 3.5e-62))
		tmp = Float64(fma(Float64(x_46_im / y_46_re), y_46_im, x_46_re) / y_46_re);
	else
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_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_] := If[Or[LessEqual[y$46$re, -2.3e+30], N[Not[LessEqual[y$46$re, 3.5e-62]], $MachinePrecision]], N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.3e30 or 3.5000000000000001e-62 < y.re

   1. Initial program 49.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 80.7

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

   5. Applied rewrites 80.7%

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

   if -2.3e30 < y.re < 3.5000000000000001e-62

   1. Initial program 70.3%

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

   3. Taylor expanded in y.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 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+30} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.re}, y.im, x.re\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-146} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5e-146) (not (<= y.re 3.5e-62)))
   (/ 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_re <= -5e-146) || !(y_46_re <= 3.5e-62)) {
		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_46re <= (-5d-146)) .or. (.not. (y_46re <= 3.5d-62))) 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_re <= -5e-146) || !(y_46_re <= 3.5e-62)) {
		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_re <= -5e-146) or not (y_46_re <= 3.5e-62):
		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_re <= -5e-146) || !(y_46_re <= 3.5e-62))
		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_re <= -5e-146) || ~((y_46_re <= 3.5e-62)))
		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[Or[LessEqual[y$46$re, -5e-146], N[Not[LessEqual[y$46$re, 3.5e-62]], $MachinePrecision]], 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.re < -4.99999999999999957e-146 or 3.5000000000000001e-62 < y.re

   1. Initial program 55.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. 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 65.3

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

   5. Applied rewrites 65.3%

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

   if -4.99999999999999957e-146 < y.re < 3.5000000000000001e-62

   1. Initial program 67.5%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 72.1

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

   5. Applied rewrites 72.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-146} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 40.1

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

5. Applied rewrites 40.1%

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

6. Final simplification 40.1%

   \[\leadsto \frac{x.im}{y.im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))