_divideComplex, imaginary part

Percentage Accurate: 61.6% → 83.7%

Time: 8.8s

Alternatives: 11

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: 61.6% 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: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, \frac{y.re \cdot x.im}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re y.re (* y.im y.im))))
   (if (<= y.im -4.4e+126)
     (/ (fma y.re (/ (fma y.re (/ x.re y.im) x.im) y.im) (- x.re)) y.im)
     (if (<= y.im -2.32e-73)
       (/ (fma x.re (- y.im) (* y.re x.im)) t_0)
       (if (<= y.im 2e-121)
         (/ (fma (* y.im x.re) (/ -1.0 y.re) x.im) y.re)
         (if (<= y.im 5e+146)
           (fma (- x.re) (/ y.im t_0) (/ (* y.re x.im) t_0))
           (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -4.4e+126) {
		tmp = fma(y_46_re, (fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -2.32e-73) {
		tmp = fma(x_46_re, -y_46_im, (y_46_re * x_46_im)) / t_0;
	} else if (y_46_im <= 2e-121) {
		tmp = fma((y_46_im * x_46_re), (-1.0 / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 5e+146) {
		tmp = fma(-x_46_re, (y_46_im / t_0), ((y_46_re * x_46_im) / t_0));
	} else {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.4e+126)
		tmp = Float64(fma(y_46_re, Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -2.32e-73)
		tmp = Float64(fma(x_46_re, Float64(-y_46_im), Float64(y_46_re * x_46_im)) / t_0);
	elseif (y_46_im <= 2e-121)
		tmp = Float64(fma(Float64(y_46_im * x_46_re), Float64(-1.0 / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 5e+146)
		tmp = fma(Float64(-x_46_re), Float64(y_46_im / t_0), Float64(Float64(y_46_re * x_46_im) / t_0));
	else
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.4e+126], N[(N[(y$46$re * N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -2.32e-73], N[(N[(x$46$re * (-y$46$im) + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 2e-121], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5e+146], N[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision] + N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -4.39999999999999997e126

   1. Initial program 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 31.4

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 31.4

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 31.4

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

   4. Applied egg-rr 31.4%

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

   5. Taylor expanded in y.im around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Simplified 82.5%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 85.8%

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

   if -4.39999999999999997e126 < y.im < -2.32e-73

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 90.4

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 90.4

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 90.4

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

   4. Applied egg-rr 90.4%

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

   if -2.32e-73 < y.im < 2e-121

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 69.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 69.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.2

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

   7. Simplified 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 95.2

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

   9. Applied egg-rr 95.2%

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

   if 2e-121 < y.im < 4.9999999999999999e146

   1. Initial program 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

   4. Applied egg-rr 82.3%

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

   if 4.9999999999999999e146 < y.im

   1. Initial program 33.0%

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

   3. Taylor expanded in y.re around 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. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 97.4

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

   5. Simplified 97.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;t\_0 \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re (/ x.im y.im) (- x.re))))
   (if (<= y.im -4.4e+126)
     (/ (fma y.re (/ (fma y.re (/ x.re y.im) x.im) y.im) (- x.re)) y.im)
     (if (<= y.im -2.32e-73)
       (/ (fma x.re (- y.im) (* y.re x.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 1.8e-120)
         (/ (fma (* y.im x.re) (/ -1.0 y.re) x.im) y.re)
         (if (<= y.im 6.8e+146)
           (* t_0 (/ y.im (fma y.im y.im (* y.re y.re))))
           (/ t_0 y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re);
	double tmp;
	if (y_46_im <= -4.4e+126) {
		tmp = fma(y_46_re, (fma(y_46_re, (x_46_re / y_46_im), x_46_im) / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -2.32e-73) {
		tmp = fma(x_46_re, -y_46_im, (y_46_re * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 1.8e-120) {
		tmp = fma((y_46_im * x_46_re), (-1.0 / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 6.8e+146) {
		tmp = t_0 * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} else {
		tmp = t_0 / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re))
	tmp = 0.0
	if (y_46_im <= -4.4e+126)
		tmp = Float64(fma(y_46_re, Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -2.32e-73)
		tmp = Float64(fma(x_46_re, Float64(-y_46_im), Float64(y_46_re * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.8e-120)
		tmp = Float64(fma(Float64(y_46_im * x_46_re), Float64(-1.0 / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 6.8e+146)
		tmp = Float64(t_0 * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(t_0 / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision]}, If[LessEqual[y$46$im, -4.4e+126], N[(N[(y$46$re * N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -2.32e-73], N[(N[(x$46$re * (-y$46$im) + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.8e-120], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.8e+146], N[(t$95$0 * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -4.39999999999999997e126

   1. Initial program 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 31.4

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 31.4

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 31.4

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

   4. Applied egg-rr 31.4%

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

   5. Taylor expanded in y.im around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   7. Simplified 82.5%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 85.8%

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

   if -4.39999999999999997e126 < y.im < -2.32e-73

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 90.4

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 90.4

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 90.4

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

   4. Applied egg-rr 90.4%

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

   if -2.32e-73 < y.im < 1.8000000000000001e-120

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 69.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 69.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.2

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

   7. Simplified 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 95.2

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

   9. Applied egg-rr 95.2%

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

   if 1.8000000000000001e-120 < y.im < 6.79999999999999981e146

   1. Initial program 74.7%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.8

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

   5. Simplified 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.4

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 80.4

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

   7. Applied egg-rr 80.4%

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

   if 6.79999999999999981e146 < y.im

   1. Initial program 33.0%

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

   3. Taylor expanded in y.re around 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. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 97.4

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

   5. Simplified 97.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)\\ t_1 := \frac{t\_0}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;t\_0 \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.re (/ x.im y.im) (- x.re))) (t_1 (/ t_0 y.im)))
   (if (<= y.im -4.2e+126)
     t_1
     (if (<= y.im -2.32e-73)
       (/ (fma x.re (- y.im) (* y.re x.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 1.8e-120)
         (/ (fma (* y.im x.re) (/ -1.0 y.re) x.im) y.re)
         (if (<= y.im 6.8e+146)
           (* t_0 (/ y.im (fma y.im y.im (* y.re y.re))))
           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_re, (x_46_im / y_46_im), -x_46_re);
	double t_1 = t_0 / y_46_im;
	double tmp;
	if (y_46_im <= -4.2e+126) {
		tmp = t_1;
	} else if (y_46_im <= -2.32e-73) {
		tmp = fma(x_46_re, -y_46_im, (y_46_re * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 1.8e-120) {
		tmp = fma((y_46_im * x_46_re), (-1.0 / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 6.8e+146) {
		tmp = t_0 * (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re)));
	} 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_re, Float64(x_46_im / y_46_im), Float64(-x_46_re))
	t_1 = Float64(t_0 / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.2e+126)
		tmp = t_1;
	elseif (y_46_im <= -2.32e-73)
		tmp = Float64(fma(x_46_re, Float64(-y_46_im), Float64(y_46_re * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.8e-120)
		tmp = Float64(fma(Float64(y_46_im * x_46_re), Float64(-1.0 / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 6.8e+146)
		tmp = Float64(t_0 * Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.2e+126], t$95$1, If[LessEqual[y$46$im, -2.32e-73], N[(N[(x$46$re * (-y$46$im) + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.8e-120], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.8e+146], N[(t$95$0 * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.1999999999999998e126 or 6.79999999999999981e146 < y.im

   1. Initial program 32.3%

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

   3. Taylor expanded in y.re around 0

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 92.2

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

   5. Simplified 92.2%

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

   if -4.1999999999999998e126 < y.im < -2.32e-73

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 90.4

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 90.4

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 90.4

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

   4. Applied egg-rr 90.4%

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

   if -2.32e-73 < y.im < 1.8000000000000001e-120

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 69.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 69.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.2

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

   7. Simplified 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 95.2

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

   9. Applied egg-rr 95.2%

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

   if 1.8000000000000001e-120 < y.im < 6.79999999999999981e146

   1. Initial program 74.7%

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

   3. Taylor expanded in y.im around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.8

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

   5. Simplified 72.8%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.4

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 80.4

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

   7. Applied egg-rr 80.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right) \cdot \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (fma x.re (- y.im) (* y.re x.im)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -4.2e+126)
     t_1
     (if (<= y.im -2.32e-73)
       t_0
       (if (<= y.im 2e-121)
         (/ (fma (* y.im x.re) (/ -1.0 y.re) x.im) y.re)
         (if (<= y.im 3.8e+38) 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 = fma(x_46_re, -y_46_im, (y_46_re * x_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	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 <= -4.2e+126) {
		tmp = t_1;
	} else if (y_46_im <= -2.32e-73) {
		tmp = t_0;
	} else if (y_46_im <= 2e-121) {
		tmp = fma((y_46_im * x_46_re), (-1.0 / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 3.8e+38) {
		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(fma(x_46_re, Float64(-y_46_im), Float64(y_46_re * x_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	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 <= -4.2e+126)
		tmp = t_1;
	elseif (y_46_im <= -2.32e-73)
		tmp = t_0;
	elseif (y_46_im <= 2e-121)
		tmp = Float64(fma(Float64(y_46_im * x_46_re), Float64(-1.0 / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 3.8e+38)
		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$46$re * (-y$46$im) + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $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, -4.2e+126], t$95$1, If[LessEqual[y$46$im, -2.32e-73], t$95$0, If[LessEqual[y$46$im, 2e-121], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1999999999999998e126 or 3.7999999999999998e38 < y.im

   1. Initial program 41.2%

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.8

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

   5. Simplified 88.8%

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

   if -4.1999999999999998e126 < y.im < -2.32e-73 or 2e-121 < y.im < 3.7999999999999998e38

   1. Initial program 85.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 85.8

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 85.8

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 85.9

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

   4. Applied egg-rr 85.9%

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

   if -2.32e-73 < y.im < 2e-121

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 69.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 69.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 95.2

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

   7. Simplified 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 95.2

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

   9. Applied egg-rr 95.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -260000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -260000000000.0)
     t_0
     (if (<= y.im 2.4e-81)
       (/ (fma (* y.im x.re) (/ -1.0 y.re) x.im) 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 <= -260000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.4e-81) {
		tmp = fma((y_46_im * x_46_re), (-1.0 / y_46_re), x_46_im) / 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 <= -260000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.4e-81)
		tmp = Float64(fma(Float64(y_46_im * x_46_re), Float64(-1.0 / y_46_re), x_46_im) / 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, -260000000000.0], t$95$0, If[LessEqual[y$46$im, 2.4e-81], N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.6e11 or 2.3999999999999999e-81 < y.im

   1. Initial program 56.3%

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

   3. Taylor expanded in y.re around 0

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.0

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

   5. Simplified 82.0%

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

   if -2.6e11 < y.im < 2.3999999999999999e-81

   1. Initial program 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 72.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 72.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 72.7

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

   4. Applied egg-rr 72.7%

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

   5. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 88.3

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

   7. Simplified 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. remove-double-neg N/A

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

      15. lower-/.f64 88.3

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

   9. Applied egg-rr 88.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re \cdot y.im, \frac{-1}{y.re}, x.im\right)}}{y.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -260000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im \cdot x.re, \frac{-1}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -260000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -260000000000.0)
     t_0
     (if (<= y.im 2.4e-81) (/ (- x.im (/ (* y.im x.re) 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 <= -260000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.4e-81) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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 <= -260000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.4e-81)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / 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, -260000000000.0], t$95$0, If[LessEqual[y$46$im, 2.4e-81], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.6e11 or 2.3999999999999999e-81 < y.im

   1. Initial program 56.3%

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

   3. Taylor expanded in y.re around 0

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.0

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

   5. Simplified 82.0%

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

   if -2.6e11 < y.im < 2.3999999999999999e-81

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 88.3

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

   5. Simplified 88.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -260000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -300000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -300000000000.0)
     t_0
     (if (<= y.im 5e+36) (/ (- x.im (/ (* y.im x.re) 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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -300000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 5e+36) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	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_46re / -y_46im
    if (y_46im <= (-300000000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 5d+36) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    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_re / -y_46_im;
	double tmp;
	if (y_46_im <= -300000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 5e+36) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -300000000000.0:
		tmp = t_0
	elif y_46_im <= 5e+36:
		tmp = (x_46_im - ((y_46_im * x_46_re) / 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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -300000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 5e+36)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	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_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -300000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 5e+36)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -300000000000.0], t$95$0, If[LessEqual[y$46$im, 5e+36], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3e11 or 4.99999999999999977e36 < y.im

   1. Initial program 51.7%

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

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

   5. Simplified 80.7%

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

   if -3e11 < y.im < 4.99999999999999977e36

   1. Initial program 74.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 82.6

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

   5. Simplified 82.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -300000000000:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;y.im \cdot \frac{-x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -7.2e+118)
     t_0
     (if (<= y.im -1.2e-67)
       (* y.im (/ (- x.re) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 2.3e-104) (/ x.im 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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -7.2e+118) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-67) {
		tmp = y_46_im * (-x_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_im <= 2.3e-104) {
		tmp = x_46_im / 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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -7.2e+118)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-67)
		tmp = Float64(y_46_im * Float64(Float64(-x_46_re) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 2.3e-104)
		tmp = Float64(x_46_im / 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[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -7.2e+118], t$95$0, If[LessEqual[y$46$im, -1.2e-67], N[(y$46$im * N[((-x$46$re) / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3e-104], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.2e118 or 2.2999999999999999e-104 < y.im

   1. Initial program 50.5%

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

   3. Taylor expanded in y.re around 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 75.4

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

   5. Simplified 75.4%

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

   if -7.2e118 < y.im < -1.2e-67

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 89.7

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 89.7

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 89.7

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

   4. Applied egg-rr 89.7%

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

   5. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.9

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

   7. Simplified 70.9%

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

   if -1.2e-67 < y.im < 2.2999999999999999e-104

   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 inf

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

      1. lower-/.f64 74.2

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

   5. Simplified 74.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;y.im \cdot \frac{-x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -3.4e+129)
     t_0
     (if (<= y.im -1.2e-67)
       (* x.re (/ (- y.im) (fma y.re y.re (* y.im y.im))))
       (if (<= y.im 2.3e-104) (/ x.im 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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -3.4e+129) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-67) {
		tmp = x_46_re * (-y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im)));
	} else if (y_46_im <= 2.3e-104) {
		tmp = x_46_im / 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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.4e+129)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-67)
		tmp = Float64(x_46_re * Float64(Float64(-y_46_im) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 2.3e-104)
		tmp = Float64(x_46_im / 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[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -3.4e+129], t$95$0, If[LessEqual[y$46$im, -1.2e-67], N[(x$46$re * N[((-y$46$im) / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3e-104], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.40000000000000018e129 or 2.2999999999999999e-104 < y.im

   1. Initial program 50.1%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 75.6

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

   5. Simplified 75.6%

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

   if -3.40000000000000018e129 < y.im < -1.2e-67

   1. Initial program 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. div-inv N/A

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

      8. lift-+.f64 N/A

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

      9. flip-+ N/A

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

      10. clear-num N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 N/A

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 69.9

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

   7. Simplified 69.9%

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

   if -1.2e-67 < y.im < 2.2999999999999999e-104

   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 inf

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

      1. lower-/.f64 74.2

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

   5. Simplified 74.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;x.re \cdot \frac{-y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -80000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -80000000000.0)
     t_0
     (if (<= y.im 2.3e-104) (/ x.im 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 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -80000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-104) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	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_46re / -y_46im
    if (y_46im <= (-80000000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 2.3d-104) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    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_re / -y_46_im;
	double tmp;
	if (y_46_im <= -80000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-104) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -80000000000.0:
		tmp = t_0
	elif y_46_im <= 2.3e-104:
		tmp = x_46_im / 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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -80000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-104)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	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_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -80000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-104)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -80000000000.0], t$95$0, If[LessEqual[y$46$im, 2.3e-104], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8e10 or 2.2999999999999999e-104 < y.im

   1. Initial program 56.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 74.6

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

   5. Simplified 74.6%

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

   if -8e10 < y.im < 2.2999999999999999e-104

   1. Initial program 73.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 69.5

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

   5. Simplified 69.5%

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

Alternative 11: 43.0% 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 64.0%

   \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 41.3

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

5. Simplified 41.3%

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

Reproduce

herbie shell --seed 2024219 
(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))))