_divideComplex, imaginary part

Percentage Accurate: 61.8% → 82.1%

Time: 8.4s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.32 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{t\_0}, x.im, \frac{x.re}{t\_0} \cdot \left(-y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma (/ 1.0 y.re) x.im (* (/ (- y.im) y.re) (/ x.re y.re)))))
   (if (<= y.re -4.8e+126)
     t_1
     (if (<= y.re -1.05e-160)
       (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.re 2.1e-6)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.32e+113)
           (fma (/ y.re t_0) x.im (* (/ x.re t_0) (- y.im)))
           t_1))))))

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, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((1.0 / y_46_re), x_46_im, ((-y_46_im / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -4.8e+126) {
		tmp = t_1;
	} else if (y_46_re <= -1.05e-160) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_re <= 2.1e-6) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.32e+113) {
		tmp = fma((y_46_re / t_0), x_46_im, ((x_46_re / t_0) * -y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(1.0 / y_46_re), x_46_im, Float64(Float64(Float64(-y_46_im) / y_46_re) * Float64(x_46_re / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.8e+126)
		tmp = t_1;
	elseif (y_46_re <= -1.05e-160)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 2.1e-6)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.32e+113)
		tmp = fma(Float64(y_46_re / t_0), x_46_im, Float64(Float64(x_46_re / t_0) * Float64(-y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$re), $MachinePrecision] * x$46$im + N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.8e+126], t$95$1, If[LessEqual[y$46$re, -1.05e-160], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-6], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.32e+113], N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im + N[(N[(x$46$re / t$95$0), $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.80000000000000024e126 or 1.31999999999999996e113 < y.re

   1. Initial program 42.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

   8. Taylor expanded in y.re around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{y.re}, x.im, \color{blue}{\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 92.5

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

   10. Applied rewrites 92.5%

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

   if -4.80000000000000024e126 < y.re < -1.05e-160

   1. Initial program 84.1%

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

   if -1.05e-160 < y.re < 2.0999999999999998e-6

   1. Initial program 69.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 2.0999999999999998e-6 < y.re < 1.31999999999999996e113

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 91.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.32 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.im, \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ 1.0 y.re) x.im (* (/ (- y.im) y.re) (/ x.re y.re)))))
   (if (<= y.re -4.8e+126)
     t_0
     (if (<= y.re -1.05e-160)
       (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.re 2.4e+15) (/ (- (/ (* x.im y.re) y.im) x.re) y.im) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((1.0 / y_46_re), x_46_im, ((-y_46_im / y_46_re) * (x_46_re / y_46_re)));
	double tmp;
	if (y_46_re <= -4.8e+126) {
		tmp = t_0;
	} else if (y_46_re <= -1.05e-160) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_re <= 2.4e+15) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(1.0 / y_46_re), x_46_im, Float64(Float64(Float64(-y_46_im) / y_46_re) * Float64(x_46_re / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.8e+126)
		tmp = t_0;
	elseif (y_46_re <= -1.05e-160)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 2.4e+15)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(1.0 / y$46$re), $MachinePrecision] * x$46$im + N[(N[((-y$46$im) / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.8e+126], t$95$0, If[LessEqual[y$46$re, -1.05e-160], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+15], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.80000000000000024e126 or 2.4e15 < y.re

   1. Initial program 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 59.3%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 81.1

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

   7. Applied rewrites 81.1%

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

   8. Taylor expanded in y.re around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{y.re}, x.im, \color{blue}{\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 88.5

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

   10. Applied rewrites 88.5%

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

   if -4.80000000000000024e126 < y.re < -1.05e-160

   1. Initial program 84.1%

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

   if -1.05e-160 < y.re < 2.4e15

   1. Initial program 70.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y.re}, x.im, \frac{-y.im}{y.re} \cdot \frac{x.re}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)}{-\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* x.re y.im) y.re)) y.re))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -7e-9)
     t_1
     (if (<= y.im 1.8e-119)
       t_0
       (if (<= y.im 4.5)
         (/
          (fma (- x.im) y.re (* x.re y.im))
          (- (fma y.im y.im (* y.re y.re))))
         (if (<= y.im 2.15e+51) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-9) {
		tmp = t_1;
	} else if (y_46_im <= 1.8e-119) {
		tmp = t_0;
	} else if (y_46_im <= 4.5) {
		tmp = fma(-x_46_im, y_46_re, (x_46_re * y_46_im)) / -fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 2.15e+51) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re)
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-9)
		tmp = t_1;
	elseif (y_46_im <= 1.8e-119)
		tmp = t_0;
	elseif (y_46_im <= 4.5)
		tmp = Float64(fma(Float64(-x_46_im), y_46_re, Float64(x_46_re * y_46_im)) / Float64(-fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	elseif (y_46_im <= 2.15e+51)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-9], t$95$1, If[LessEqual[y$46$im, 1.8e-119], t$95$0, If[LessEqual[y$46$im, 4.5], N[(N[((-x$46$im) * y$46$re + N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / (-N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[y$46$im, 2.15e+51], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.9999999999999998e-9 or 2.1499999999999999e51 < y.im

   1. Initial program 52.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 79.5%

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

   if -6.9999999999999998e-9 < y.im < 1.8e-119 or 4.5 < y.im < 2.1499999999999999e51

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if 1.8e-119 < y.im < 4.5

   1. Initial program 95.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 96.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 96.0

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

   4. Applied rewrites 96.0%

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

Alternative 4: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.5:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* x.re y.im) y.re)) y.re))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -7e-9)
     t_1
     (if (<= y.im 1.8e-119)
       t_0
       (if (<= y.im 4.5)
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.im y.im) (* y.re y.re)))
         (if (<= y.im 2.15e+51) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double t_1 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-9) {
		tmp = t_1;
	} else if (y_46_im <= 1.8e-119) {
		tmp = t_0;
	} else if (y_46_im <= 4.5) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 2.15e+51) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re)
	t_1 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-9)
		tmp = t_1;
	elseif (y_46_im <= 1.8e-119)
		tmp = t_0;
	elseif (y_46_im <= 4.5)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 2.15e+51)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-9], t$95$1, If[LessEqual[y$46$im, 1.8e-119], t$95$0, If[LessEqual[y$46$im, 4.5], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.15e+51], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.9999999999999998e-9 or 2.1499999999999999e51 < y.im

   1. Initial program 52.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 79.5%

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

   if -6.9999999999999998e-9 < y.im < 1.8e-119 or 4.5 < y.im < 2.1499999999999999e51

   1. Initial program 73.1%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if 1.8e-119 < y.im < 4.5

   1. Initial program 95.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.5:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -21:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -21.0)
   (/ x.im y.re)
   (if (<= y.re 2.4e+15)
     (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)
     (if (<= y.re 5.2e+98)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.re y.re))
       (/ x.im 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 <= -21.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+15) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else if (y_46_re <= 5.2e+98) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / 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 <= -21.0)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.4e+15)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 5.2e+98)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / 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, -21.0], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+15], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+98], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -21 or 5.1999999999999999e98 < y.re

   1. Initial program 51.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -21 < y.re < 2.4e15

   1. Initial program 75.4%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot x.re + \left(-1 \cdot \frac{x.im \cdot {y.re}^{3}}{{y.im}^{3}} + \frac{x.im \cdot y.re}{y.im}\right)\right) - -1 \cdot \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}} \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \mathsf{fma}\left(\frac{y.re}{y.im}, \frac{y.re}{y.im}, -1\right), x.im \cdot \left(\frac{y.re}{y.im} - {\left(\frac{y.re}{y.im}\right)}^{3}\right)\right)}{y.im}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.0%

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

   if 2.4e15 < y.re < 5.1999999999999999e98

   1. Initial program 91.5%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

Alternative 6: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.92:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, \left(-y.im\right) \cdot x.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.92)
   (/ x.im y.re)
   (if (<= y.re 9e-95)
     (/ (fma y.re x.im (* (- y.im) x.re)) (* y.im y.im))
     (if (<= y.re 5.2e+98)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.re y.re))
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9e-95) {
		tmp = fma(y_46_re, x_46_im, (-y_46_im * x_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 5.2e+98) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 9e-95)
		tmp = Float64(fma(y_46_re, x_46_im, Float64(Float64(-y_46_im) * x_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 5.2e+98)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 9e-95], N[(N[(y$46$re * x$46$im + N[((-y$46$im) * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+98], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.9199999999999999 or 5.1999999999999999e98 < y.re

   1. Initial program 51.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -1.9199999999999999 < y.re < 9e-95

   1. Initial program 77.6%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 67.6

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

   7. Applied rewrites 67.6%

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

   if 9e-95 < y.re < 5.1999999999999999e98

   1. Initial program 79.1%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.92:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, \left(-y.im\right) \cdot x.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -1.92:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-95}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.re -1.92)
     (/ x.im y.re)
     (if (<= y.re 9e-95)
       (/ t_0 (* y.im y.im))
       (if (<= y.re 5.2e+98) (/ t_0 (* y.re y.re)) (/ x.im 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_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -1.92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9e-95) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_re <= 5.2e+98) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * y_46re) - (x_46re * y_46im)
    if (y_46re <= (-1.92d0)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 9d-95) then
        tmp = t_0 / (y_46im * y_46im)
    else if (y_46re <= 5.2d+98) then
        tmp = t_0 / (y_46re * y_46re)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -1.92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9e-95) {
		tmp = t_0 / (y_46_im * y_46_im);
	} else if (y_46_re <= 5.2e+98) {
		tmp = t_0 / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if y_46_re <= -1.92:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 9e-95:
		tmp = t_0 / (y_46_im * y_46_im)
	elif y_46_re <= 5.2e+98:
		tmp = t_0 / (y_46_re * y_46_re)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 9e-95)
		tmp = Float64(t_0 / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 5.2e+98)
		tmp = Float64(t_0 / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.92)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 9e-95)
		tmp = t_0 / (y_46_im * y_46_im);
	elseif (y_46_re <= 5.2e+98)
		tmp = t_0 / (y_46_re * y_46_re);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 9e-95], N[(t$95$0 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+98], N[(t$95$0 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.9199999999999999 or 5.1999999999999999e98 < y.re

   1. Initial program 51.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -1.9199999999999999 < y.re < 9e-95

   1. Initial program 77.6%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 9e-95 < y.re < 5.1999999999999999e98

   1. Initial program 79.1%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

Alternative 8: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -7e-9)
     t_0
     (if (<= y.im 3.2e-35) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-9) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-35) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-9)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-35)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-9], t$95$0, If[LessEqual[y$46$im, 3.2e-35], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.9999999999999998e-9 or 3.1999999999999998e-35 < y.im

   1. Initial program 55.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 75.9%

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

   if -6.9999999999999998e-9 < y.im < 3.1999999999999998e-35

   1. Initial program 78.0%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 9: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)))
   (if (<= y.im -7e-9)
     t_0
     (if (<= y.im 3.1e-33) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -7e-9) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-33) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7e-9)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-33)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-9], t$95$0, If[LessEqual[y$46$im, 3.1e-33], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.9999999999999998e-9 or 3.09999999999999997e-33 < y.im

   1. Initial program 55.1%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot x.re + \left(-1 \cdot \frac{x.im \cdot {y.re}^{3}}{{y.im}^{3}} + \frac{x.im \cdot y.re}{y.im}\right)\right) - -1 \cdot \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}} \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, \mathsf{fma}\left(\frac{y.re}{y.im}, \frac{y.re}{y.im}, -1\right), x.im \cdot \left(\frac{y.re}{y.im} - {\left(\frac{y.re}{y.im}\right)}^{3}\right)\right)}{y.im}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \frac{-1 \cdot x.re + \frac{x.im \cdot y.re}{y.im}}{y.im} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.2%

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

   if -6.9999999999999998e-9 < y.im < 3.09999999999999997e-33

   1. Initial program 78.4%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 10: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.92:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.92)
   (/ x.im y.re)
   (if (<= y.re -4.5e-154)
     (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
     (if (<= y.re 2.4e+15) (/ (- x.re) y.im) (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.5e-154) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 2.4e+15) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.92d0)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-4.5d-154)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    else if (y_46re <= 2.4d+15) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.92) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -4.5e-154) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_re <= 2.4e+15) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.92:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -4.5e-154:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	elif y_46_re <= 2.4e+15:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.92)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -4.5e-154)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_re <= 2.4e+15)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.92)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -4.5e-154)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	elseif (y_46_re <= 2.4e+15)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.92], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.5e-154], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+15], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.9199999999999999 or 2.4e15 < y.re

   1. Initial program 58.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if -1.9199999999999999 < y.re < -4.4999999999999997e-154

   1. Initial program 91.1%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -4.4999999999999997e-154 < y.re < 2.4e15

   1. Initial program 70.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.0

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

   5. Applied rewrites 68.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.92:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.8e+144)
   (/ x.im y.re)
   (if (<= y.re -2.2e-77)
     (* (/ y.re (fma y.im y.im (* y.re y.re))) x.im)
     (if (<= y.re 2.4e+15) (/ (- x.re) y.im) (/ x.im 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 <= -5.8e+144) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -2.2e-77) {
		tmp = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	} else if (y_46_re <= 2.4e+15) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / 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 <= -5.8e+144)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -2.2e-77)
		tmp = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im);
	elseif (y_46_re <= 2.4e+15)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / 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, -5.8e+144], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.2e-77], N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+15], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.79999999999999996e144 or 2.4e15 < y.re

   1. Initial program 54.3%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -5.79999999999999996e144 < y.re < -2.20000000000000007e-77

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -2.20000000000000007e-77 < y.re < 2.4e15

   1. Initial program 73.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8e-77)
   (/ x.im y.re)
   (if (<= y.re 2.4e+15) (/ (- x.re) y.im) (/ x.im 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 <= -8e-77) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+15) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-8d-77)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.4d+15) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e-77) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.4e+15) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8e-77:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.4e+15:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / 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 <= -8e-77)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.4e+15)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8e-77)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.4e+15)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8e-77], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+15], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.9999999999999994e-77 or 2.4e15 < y.re

   1. Initial program 61.5%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -7.9999999999999994e-77 < y.re < 2.4e15

   1. Initial program 73.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.3% accurate, 3.2× speedup

Math

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

FPCore

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

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_re;
}

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_46re
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_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 44.4

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

5. Applied rewrites 44.4%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))