_divideComplex, imaginary part

Percentage Accurate: 61.2% → 82.8%

Time: 7.8s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-160}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{t\_0}, x.im, \frac{x.re}{t\_0} \cdot \left(-y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -4.9e+60)
     t_1
     (if (<= y.im -2.35e-160)
       (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
       (if (<= y.im 2.15e-181)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 4.8e+137)
           (fma (/ y.re t_0) x.im (* (/ x.re t_0) (- y.im)))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.9e+60) {
		tmp = t_1;
	} else if (y_46_im <= -2.35e-160) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else if (y_46_im <= 2.15e-181) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.8e+137) {
		tmp = fma((y_46_re / t_0), x_46_im, ((x_46_re / t_0) * -y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.9e+60)
		tmp = t_1;
	elseif (y_46_im <= -2.35e-160)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	elseif (y_46_im <= 2.15e-181)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4.8e+137)
		tmp = fma(Float64(y_46_re / t_0), x_46_im, Float64(Float64(x_46_re / t_0) * Float64(-y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.9e+60], t$95$1, If[LessEqual[y$46$im, -2.35e-160], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 2.15e-181], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.8e+137], N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im + N[(N[(x$46$re / t$95$0), $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.9000000000000003e60 or 4.79999999999999966e137 < y.im

   1. Initial program 36.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 86.0%

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

   if -4.9000000000000003e60 < y.im < -2.3499999999999999e-160

   1. Initial program 92.2%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.2

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

   4. Applied rewrites 92.2%

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

   if -2.3499999999999999e-160 < y.im < 2.15e-181

   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

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 92.6

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

   5. Applied rewrites 92.6%

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

   if 2.15e-181 < y.im < 4.79999999999999966e137

   1. Initial program 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 87.1%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-160}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.im, \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -4.9e+60)
     t_1
     (if (<= y.im -2.3e-154)
       (/ (fma (- y.im) x.re (* y.re x.im)) t_0)
       (if (<= y.im 4e-115)
         (/ (fma (- x.re) (/ y.im y.re) x.im) y.re)
         (if (<= y.im 7.6e+93)
           (* (/ -1.0 t_0) (fma (- x.im) y.re (* x.re y.im)))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.9e+60) {
		tmp = t_1;
	} else if (y_46_im <= -2.3e-154) {
		tmp = fma(-y_46_im, x_46_re, (y_46_re * x_46_im)) / t_0;
	} else if (y_46_im <= 4e-115) {
		tmp = fma(-x_46_re, (y_46_im / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 7.6e+93) {
		tmp = (-1.0 / t_0) * fma(-x_46_im, y_46_re, (x_46_re * y_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.9e+60)
		tmp = t_1;
	elseif (y_46_im <= -2.3e-154)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / t_0);
	elseif (y_46_im <= 4e-115)
		tmp = Float64(fma(Float64(-x_46_re), Float64(y_46_im / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 7.6e+93)
		tmp = Float64(Float64(-1.0 / t_0) * fma(Float64(-x_46_im), y_46_re, Float64(x_46_re * y_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.9e+60], t$95$1, If[LessEqual[y$46$im, -2.3e-154], N[(N[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 4e-115], N[(N[((-x$46$re) * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.6e+93], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[((-x$46$im) * y$46$re + N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.9000000000000003e60 or 7.5999999999999996e93 < y.im

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 83.8%

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

   if -4.9000000000000003e60 < y.im < -2.3e-154

   1. Initial program 92.2%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.2

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

   4. Applied rewrites 92.2%

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

   if -2.3e-154 < y.im < 4.0000000000000002e-115

   1. Initial program 73.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   7. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 92.6

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

   8. Applied rewrites 92.6%

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

   if 4.0000000000000002e-115 < y.im < 7.5999999999999996e93

   1. Initial program 87.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 87.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 87.6

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

   4. Applied rewrites 87.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (fma (- y.im) x.re (* y.re x.im)) (fma y.im y.im (* y.re y.re))))
        (t_1 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -4.9e+60)
     t_1
     (if (<= y.im -2.3e-154)
       t_0
       (if (<= y.im 4e-115)
         (/ (fma (- x.re) (/ y.im y.re) x.im) y.re)
         (if (<= y.im 7.6e+93) 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(-y_46_im, x_46_re, (y_46_re * x_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.9e+60) {
		tmp = t_1;
	} else if (y_46_im <= -2.3e-154) {
		tmp = t_0;
	} else if (y_46_im <= 4e-115) {
		tmp = fma(-x_46_re, (y_46_im / y_46_re), x_46_im) / y_46_re;
	} else if (y_46_im <= 7.6e+93) {
		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(Float64(-y_46_im), x_46_re, Float64(y_46_re * x_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	t_1 = Float64(fma(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.9e+60)
		tmp = t_1;
	elseif (y_46_im <= -2.3e-154)
		tmp = t_0;
	elseif (y_46_im <= 4e-115)
		tmp = Float64(fma(Float64(-x_46_re), Float64(y_46_im / y_46_re), x_46_im) / y_46_re);
	elseif (y_46_im <= 7.6e+93)
		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[((-y$46$im) * x$46$re + N[(y$46$re * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.9e+60], t$95$1, If[LessEqual[y$46$im, -2.3e-154], t$95$0, If[LessEqual[y$46$im, 4e-115], N[(N[((-x$46$re) * N[(y$46$im / y$46$re), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7.6e+93], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.9000000000000003e60 or 7.5999999999999996e93 < y.im

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   7. Applied rewrites 83.8%

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

   if -4.9000000000000003e60 < y.im < -2.3e-154 or 4.0000000000000002e-115 < y.im < 7.5999999999999996e93

   1. Initial program 89.8%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 89.8

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 89.8

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

   4. Applied rewrites 89.8%

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

   if -2.3e-154 < y.im < 4.0000000000000002e-115

   1. Initial program 73.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   7. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 92.6

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

   8. Applied rewrites 92.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 340000000:\\ \;\;\;\;\frac{y.im}{t\_1} \cdot \left(-x.re\right)\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im}{t\_1} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (fma y.im y.im (* y.re y.re))))
   (if (<= y.im -9.5e+29)
     t_0
     (if (<= y.im -1.12e-153)
       (/ (- (* y.re x.im) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 7.7e-115)
         (/ x.im y.re)
         (if (<= y.im 340000000.0)
           (* (/ y.im t_1) (- x.re))
           (if (<= y.im 7.5e+41) (* (/ x.im t_1) 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 t_1 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -9.5e+29) {
		tmp = t_0;
	} else if (y_46_im <= -1.12e-153) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 7.7e-115) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 340000000.0) {
		tmp = (y_46_im / t_1) * -x_46_re;
	} else if (y_46_im <= 7.5e+41) {
		tmp = (x_46_im / t_1) * 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(Float64(-x_46_re) / y_46_im)
	t_1 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -9.5e+29)
		tmp = t_0;
	elseif (y_46_im <= -1.12e-153)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 7.7e-115)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 340000000.0)
		tmp = Float64(Float64(y_46_im / t_1) * Float64(-x_46_re));
	elseif (y_46_im <= 7.5e+41)
		tmp = Float64(Float64(x_46_im / t_1) * 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]}, Block[{t$95$1 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+29], t$95$0, If[LessEqual[y$46$im, -1.12e-153], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.7e-115], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 340000000.0], N[(N[(y$46$im / t$95$1), $MachinePrecision] * (-x$46$re)), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+41], N[(N[(x$46$im / t$95$1), $MachinePrecision] * y$46$re), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.5000000000000003e29 or 7.50000000000000072e41 < y.im

   1. Initial program 43.2%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   if -9.5000000000000003e29 < y.im < -1.12000000000000005e-153

   1. Initial program 93.9%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if -1.12000000000000005e-153 < y.im < 7.7000000000000002e-115

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   if 7.7000000000000002e-115 < y.im < 3.4e8

   1. Initial program 95.0%

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

   3. Taylor expanded in x.re around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 3.4e8 < y.im < 7.50000000000000072e41

   1. Initial program 73.1%

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

   3. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{-153}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 340000000:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := y.re \cdot x.im - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{t\_1}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;\frac{t\_1}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (- (* y.re x.im) (* x.re y.im))))
   (if (<= y.im -9.5e+29)
     t_0
     (if (<= y.im -3.2e-72)
       (/ t_1 (* y.im y.im))
       (if (<= y.im 1.15e-34)
         (/ t_1 (* y.re y.re))
         (if (<= y.im 7.5e+41)
           (* (/ y.re (fma y.re y.re (* y.im y.im))) x.im)
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_im <= -9.5e+29) {
		tmp = t_0;
	} else if (y_46_im <= -3.2e-72) {
		tmp = t_1 / (y_46_im * y_46_im);
	} else if (y_46_im <= 1.15e-34) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 7.5e+41) {
		tmp = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_im <= -9.5e+29)
		tmp = t_0;
	elseif (y_46_im <= -3.2e-72)
		tmp = Float64(t_1 / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 1.15e-34)
		tmp = Float64(t_1 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 7.5e+41)
		tmp = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+29], t$95$0, If[LessEqual[y$46$im, -3.2e-72], N[(t$95$1 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.15e-34], N[(t$95$1 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.5e+41], N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.5000000000000003e29 or 7.50000000000000072e41 < y.im

   1. Initial program 43.2%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 73.9

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

   5. Applied rewrites 73.9%

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

   if -9.5000000000000003e29 < y.im < -3.19999999999999999e-72

   1. Initial program 88.8%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -3.19999999999999999e-72 < y.im < 1.15000000000000006e-34

   1. Initial program 80.9%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if 1.15000000000000006e-34 < y.im < 7.50000000000000072e41

   1. Initial program 76.6%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 76.6

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -9.5e+29)
     t_0
     (if (<= y.im -9.6e-72)
       (/ (- (* y.re x.im) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 2.3e+53) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -9.5e+29) {
		tmp = t_0;
	} else if (y_46_im <= -9.6e-72) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.3e+53) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 <= (-9.5d+29)) then
        tmp = t_0
    else if (y_46im <= (-9.6d-72)) then
        tmp = ((y_46re * x_46im) - (x_46re * y_46im)) / (y_46im * y_46im)
    else if (y_46im <= 2.3d+53) then
        tmp = (x_46im - ((x_46re * y_46im) / 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 <= -9.5e+29) {
		tmp = t_0;
	} else if (y_46_im <= -9.6e-72) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.3e+53) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 <= -9.5e+29:
		tmp = t_0
	elif y_46_im <= -9.6e-72:
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	elif y_46_im <= 2.3e+53:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.5e+29)
		tmp = t_0;
	elseif (y_46_im <= -9.6e-72)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.3e+53)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

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 <= -9.5e+29)
		tmp = t_0;
	elseif (y_46_im <= -9.6e-72)
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	elseif (y_46_im <= 2.3e+53)
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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, -9.5e+29], t$95$0, If[LessEqual[y$46$im, -9.6e-72], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.3e+53], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.5000000000000003e29 or 2.3000000000000002e53 < y.im

   1. Initial program 43.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 74.3

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

   5. Applied rewrites 74.3%

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

   if -9.5000000000000003e29 < y.im < -9.6e-72

   1. Initial program 88.8%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -9.6e-72 < y.im < 2.3000000000000002e53

   1. Initial program 79.8%

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

   3. 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 79.2

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

   5. Applied rewrites 79.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.7e-22)
   (/ x.im y.re)
   (if (<= y.re 1.75e-82)
     (/ (- x.re) y.im)
     (if (<= y.re 1.1e+38)
       (/ (* y.re x.im) (fma y.im y.im (* y.re y.re)))
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.7e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.75e-82) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.1e+38) {
		tmp = (y_46_re * x_46_im) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.7e-22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.75e-82)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.1e+38)
		tmp = Float64(Float64(y_46_re * x_46_im) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.7e-22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.75e-82], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.1e+38], N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.6999999999999999e-22 or 1.10000000000000003e38 < y.re

   1. Initial program 46.0%

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

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

   5. Applied rewrites 65.7%

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

   if -1.6999999999999999e-22 < y.re < 1.7499999999999999e-82

   1. Initial program 74.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

   if 1.7499999999999999e-82 < y.re < 1.10000000000000003e38

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 89.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 89.0

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

   4. Applied rewrites 89.0%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.7e-22)
   (/ x.im y.re)
   (if (<= y.re 1.75e-82)
     (/ (- x.re) y.im)
     (if (<= y.re 1.35e+99)
       (* (/ y.re (fma y.re y.re (* y.im y.im))) x.im)
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.7e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.75e-82) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.35e+99) {
		tmp = (y_46_re / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.7e-22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.75e-82)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.35e+99)
		tmp = Float64(Float64(y_46_re / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.7e-22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.75e-82], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.35e+99], N[(N[(y$46$re / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.6999999999999999e-22 or 1.34999999999999994e99 < y.re

   1. Initial program 44.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 65.8

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

   5. Applied rewrites 65.8%

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

   if -1.6999999999999999e-22 < y.re < 1.7499999999999999e-82

   1. Initial program 74.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

   if 1.7499999999999999e-82 < y.re < 1.34999999999999994e99

   1. Initial program 82.7%

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

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 82.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.9

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

   7. Applied rewrites 61.9%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{x.im}{y.im}, y.re, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ x.im y.im) y.re (- x.re)) y.im)))
   (if (<= y.im -1.05e-71)
     t_0
     (if (<= y.im 1.45e+62) (/ (fma (- x.re) (/ y.im 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((x_46_im / y_46_im), y_46_re, -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.05e-71) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e+62) {
		tmp = fma(-x_46_re, (y_46_im / 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(Float64(x_46_im / y_46_im), y_46_re, Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.05e-71)
		tmp = t_0;
	elseif (y_46_im <= 1.45e+62)
		tmp = Float64(fma(Float64(-x_46_re), Float64(y_46_im / 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[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e-71], t$95$0, If[LessEqual[y$46$im, 1.45e+62], N[(N[((-x$46$re) * N[(y$46$im / 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 < -1.0500000000000001e-71 or 1.44999999999999992e62 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 80.5%

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

   if -1.0500000000000001e-71 < y.im < 1.44999999999999992e62

   1. Initial program 79.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   7. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 79.4

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

   8. Applied rewrites 79.4%

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

Alternative 10: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* y.re x.im) y.im) x.re) y.im)))
   (if (<= y.im -9.6e-72)
     t_0
     (if (<= y.im 1.45e+62) (/ (fma (- x.re) (/ y.im 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -9.6e-72) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e+62) {
		tmp = fma(-x_46_re, (y_46_im / 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(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.6e-72)
		tmp = t_0;
	elseif (y_46_im <= 1.45e+62)
		tmp = Float64(fma(Float64(-x_46_re), Float64(y_46_im / 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[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.6e-72], t$95$0, If[LessEqual[y$46$im, 1.45e+62], N[(N[((-x$46$re) * N[(y$46$im / 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 < -9.6e-72 or 1.44999999999999992e62 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -9.6e-72 < y.im < 1.44999999999999992e62

   1. Initial program 79.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 36.9

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   7. 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. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 79.4

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

   8. Applied rewrites 79.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, \frac{y.im}{y.re}, x.im\right)}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* y.re x.im) y.im) x.re) y.im)))
   (if (<= y.im -9.6e-72)
     t_0
     (if (<= y.im 1.45e+62) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -9.6e-72) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e+62) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    if (y_46im <= (-9.6d-72)) then
        tmp = t_0
    else if (y_46im <= 1.45d+62) then
        tmp = (x_46im - ((x_46re * y_46im) / 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -9.6e-72) {
		tmp = t_0;
	} else if (y_46_im <= 1.45e+62) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -9.6e-72:
		tmp = t_0
	elif y_46_im <= 1.45e+62:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -9.6e-72)
		tmp = t_0;
	elseif (y_46_im <= 1.45e+62)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -9.6e-72)
		tmp = t_0;
	elseif (y_46_im <= 1.45e+62)
		tmp = (x_46_im - ((x_46_re * y_46_im) / 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[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -9.6e-72], t$95$0, If[LessEqual[y$46$im, 1.45e+62], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.6e-72 or 1.44999999999999992e62 < y.im

   1. Initial program 48.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -9.6e-72 < y.im < 1.44999999999999992e62

   1. Initial program 79.9%

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

   3. 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 79.4

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

   5. Applied rewrites 79.4%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 205000:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.7e-22)
   (/ x.im y.re)
   (if (<= y.re 205000.0) (/ (- x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.7e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 205000.0) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.7d-22)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 205000.0d0) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.7e-22) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 205000.0) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.7e-22:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 205000.0:
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.7e-22)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 205000.0)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.7e-22)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 205000.0)
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.7e-22], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 205000.0], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.6999999999999999e-22 or 205000 < y.re

   1. Initial program 50.4%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -1.6999999999999999e-22 < y.re < 205000

   1. Initial program 76.1%

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

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

   5. Applied rewrites 66.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 205000:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.1% 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 63.9%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 39.6

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

5. Applied rewrites 39.6%

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

Reproduce

herbie shell --seed 2024332 
(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))))