_divideComplex, imaginary part

Percentage Accurate: 61.6% → 83.9%

Time: 6.5s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{y.re}{t\_0}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x.im, \left(-y.im\right) \cdot \frac{x.re}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, t\_1 \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))) (t_1 (/ y.re t_0)))
   (if (<= y.re -8e+107)
     (/ (fma (/ y.im y.re) x.re (- x.im)) (- y.re))
     (if (<= y.re -2.1e-53)
       (fma t_1 x.im (* (- y.im) (/ x.re t_0)))
       (if (<= y.re 6.8e-67)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 4.3e+108)
           (fma (- x.re) (/ y.im t_0) (* t_1 x.im))
           (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = y_46_re / t_0;
	double tmp;
	if (y_46_re <= -8e+107) {
		tmp = fma((y_46_im / y_46_re), x_46_re, -x_46_im) / -y_46_re;
	} else if (y_46_re <= -2.1e-53) {
		tmp = fma(t_1, x_46_im, (-y_46_im * (x_46_re / t_0)));
	} else if (y_46_re <= 6.8e-67) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 4.3e+108) {
		tmp = fma(-x_46_re, (y_46_im / t_0), (t_1 * x_46_im));
	} else {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(y_46_re / t_0)
	tmp = 0.0
	if (y_46_re <= -8e+107)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_re, Float64(-x_46_im)) / Float64(-y_46_re));
	elseif (y_46_re <= -2.1e-53)
		tmp = fma(t_1, x_46_im, Float64(Float64(-y_46_im) * Float64(x_46_re / t_0)));
	elseif (y_46_re <= 6.8e-67)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 4.3e+108)
		tmp = fma(Float64(-x_46_re), Float64(y_46_im / t_0), Float64(t_1 * x_46_im));
	else
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re / t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -8e+107], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], If[LessEqual[y$46$re, -2.1e-53], N[(t$95$1 * x$46$im + N[((-y$46$im) * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.8e-67], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.3e+108], N[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision] + N[(t$95$1 * x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -7.9999999999999998e107

   1. Initial program 37.8%

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

   3. Taylor expanded in y.re around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -7.9999999999999998e107 < y.re < -2.09999999999999977e-53

   1. Initial program 76.8%

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

   4. Applied rewrites 80.6%

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

   if -2.09999999999999977e-53 < y.re < 6.8000000000000002e-67

   1. Initial program 63.8%

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

   3. Taylor expanded in y.re around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 6.8000000000000002e-67 < y.re < 4.29999999999999996e108

   1. Initial program 87.5%

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

   4. Applied rewrites 96.6%

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

   if 4.29999999999999996e108 < y.re

   1. Initial program 27.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 77.1

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

   7. Applied rewrites 77.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, x.im, \left(-y.im\right) \cdot \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, \frac{y.re}{t\_0} \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma (- x.re) (/ y.im t_0) (* (/ y.re t_0) x.im))))
   (if (<= y.re -6.4e+105)
     (/ (fma (/ y.im y.re) x.re (- x.im)) (- y.re))
     (if (<= y.re -2.1e-53)
       t_1
       (if (<= y.re 6.8e-67)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 4.3e+108)
           t_1
           (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(-x_46_re, (y_46_im / t_0), ((y_46_re / t_0) * x_46_im));
	double tmp;
	if (y_46_re <= -6.4e+105) {
		tmp = fma((y_46_im / y_46_re), x_46_re, -x_46_im) / -y_46_re;
	} else if (y_46_re <= -2.1e-53) {
		tmp = t_1;
	} else if (y_46_re <= 6.8e-67) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 4.3e+108) {
		tmp = t_1;
	} else {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(Float64(-x_46_re), Float64(y_46_im / t_0), Float64(Float64(y_46_re / t_0) * x_46_im))
	tmp = 0.0
	if (y_46_re <= -6.4e+105)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_re, Float64(-x_46_im)) / Float64(-y_46_re));
	elseif (y_46_re <= -2.1e-53)
		tmp = t_1;
	elseif (y_46_re <= 6.8e-67)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 4.3e+108)
		tmp = t_1;
	else
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x$46$re) * N[(y$46$im / t$95$0), $MachinePrecision] + N[(N[(y$46$re / t$95$0), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.4e+105], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], If[LessEqual[y$46$re, -2.1e-53], t$95$1, If[LessEqual[y$46$re, 6.8e-67], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.3e+108], t$95$1, N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.4e105

   1. Initial program 37.8%

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

   3. Taylor expanded in y.re around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -6.4e105 < y.re < -2.09999999999999977e-53 or 6.8000000000000002e-67 < y.re < 4.29999999999999996e108

   1. Initial program 82.3%

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

   4. Applied rewrites 88.1%

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

   if -2.09999999999999977e-53 < y.re < 6.8000000000000002e-67

   1. Initial program 63.8%

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

   3. Taylor expanded in y.re around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 4.29999999999999996e108 < y.re

   1. Initial program 27.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 77.1

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

   7. Applied rewrites 77.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma (/ y.im y.re) x.re (- x.im)) (- y.re))))
   (if (<= y.re -3.05e+77)
     t_0
     (if (<= y.re 3.4e-67)
       (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
       (if (<= y.re 6e+26)
         (/ (fma (- x.re) y.im (* x.im y.re)) (fma y.im y.im (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((y_46_im / y_46_re), x_46_re, -x_46_im) / -y_46_re;
	double tmp;
	if (y_46_re <= -3.05e+77) {
		tmp = t_0;
	} else if (y_46_re <= 3.4e-67) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6e+26) {
		tmp = fma(-x_46_re, y_46_im, (x_46_im * y_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(y_46_im / y_46_re), x_46_re, Float64(-x_46_im)) / Float64(-y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.05e+77)
		tmp = t_0;
	elseif (y_46_re <= 3.4e-67)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6e+26)
		tmp = Float64(fma(Float64(-x_46_re), y_46_im, Float64(x_46_im * y_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]}, If[LessEqual[y$46$re, -3.05e+77], t$95$0, If[LessEqual[y$46$re, 3.4e-67], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6e+26], N[(N[((-x$46$re) * y$46$im + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.05000000000000016e77 or 5.99999999999999994e26 < y.re

   1. Initial program 42.1%

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

   3. Taylor expanded in y.re around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 12.8

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

   5. Applied rewrites 12.8%

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

   7. Applied rewrites 18.6%

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

   8. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 84.2

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

   10. Applied rewrites 84.2%

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

   if -3.05000000000000016e77 < y.re < 3.4000000000000001e-67

   1. Initial program 65.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 84.4%

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

   if 3.4000000000000001e-67 < y.re < 5.99999999999999994e26

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.re, y.im, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -5.8e+152)
     t_0
     (if (<= y.im -5.1e-48)
       (/ (fma (- y.im) x.re (* x.im y.re)) (* y.im y.im))
       (if (<= y.im 5.5e+25) (/ (- 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 <= -5.8e+152) {
		tmp = t_0;
	} else if (y_46_im <= -5.1e-48) {
		tmp = fma(-y_46_im, x_46_re, (x_46_im * y_46_re)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 5.5e+25) {
		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 <= -5.8e+152)
		tmp = t_0;
	elseif (y_46_im <= -5.1e-48)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 5.5e+25)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -5.8e+152], t$95$0, If[LessEqual[y$46$im, -5.1e-48], N[(N[((-y$46$im) * x$46$re + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.5e+25], 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 < -5.7999999999999997e152 or 5.50000000000000018e25 < y.im

   1. Initial program 35.7%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -5.7999999999999997e152 < y.im < -5.10000000000000011e-48

   1. Initial program 86.4%

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

   4. Applied rewrites 87.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-neg.f64 N/A

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

      11. div-add-rev N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 86.4%

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

   7. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.5

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

   9. Applied rewrites 77.5%

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

   if -5.10000000000000011e-48 < y.im < 5.50000000000000018e25

   1. Initial program 65.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.16e+48)
   (/ x.im y.re)
   (if (<= y.re 8.5e-55)
     (/ (- x.re) y.im)
     (if (<= y.re 2.1e+31)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.re y.re))
       (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e+48) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 8.5e-55) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.1e+31) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

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 <= (-1.16d+48)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 8.5d-55) then
        tmp = -x_46re / y_46im
    else if (y_46re <= 2.1d+31) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46re * y_46re)
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.16e+48) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 8.5e-55) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 2.1e+31) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.16e+48:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 8.5e-55:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 2.1e+31:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re)
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.16e+48)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 8.5e-55)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 2.1e+31)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_re * y_46_re));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.16e+48)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 8.5e-55)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 2.1e+31)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.16e+48], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 8.5e-55], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.1e+31], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15999999999999992e48 or 2.09999999999999979e31 < y.re

   1. Initial program 41.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   if -1.15999999999999992e48 < y.re < 8.49999999999999968e-55

   1. Initial program 65.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   if 8.49999999999999968e-55 < y.re < 2.09999999999999979e31

   1. Initial program 99.2%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{{y.re}^{2}}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.05e+77) (not (<= y.re 9e-55)))
   (/ (fma (/ y.im y.re) x.re (- x.im)) (- y.re))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55)) {
		tmp = fma((y_46_im / y_46_re), x_46_re, -x_46_im) / -y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55))
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_re, Float64(-x_46_im)) / Float64(-y_46_re));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.05e+77], N[Not[LessEqual[y$46$re, 9e-55]], $MachinePrecision]], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$re + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.05000000000000016e77 or 8.99999999999999941e-55 < y.re

   1. Initial program 49.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 17.9

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

   5. Applied rewrites 17.9%

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

   7. Applied rewrites 22.9%

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

   8. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 81.5

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

   10. Applied rewrites 81.5%

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

   if -3.05000000000000016e77 < y.re < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.re, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.05e+77) (not (<= y.re 9e-55)))
   (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55)) {
		tmp = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55))
		tmp = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.05e+77], N[Not[LessEqual[y$46$re, 9e-55]], $MachinePrecision]], N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.05000000000000016e77 or 8.99999999999999941e-55 < y.re

   1. Initial program 49.7%

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in y.re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if -3.05000000000000016e77 < y.re < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.05e+77) (not (<= y.re 9e-55)))
   (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55)) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.05d+77)) .or. (.not. (y_46re <= 9d-55))) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55)) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.05e+77) or not (y_46_re <= 9e-55):
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.05e+77) || !(y_46_re <= 9e-55))
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.05e+77) || ~((y_46_re <= 9e-55)))
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.05e+77], N[Not[LessEqual[y$46$re, 9e-55]], $MachinePrecision]], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.05000000000000016e77 or 8.99999999999999941e-55 < y.re

   1. Initial program 49.7%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -3.05000000000000016e77 < y.re < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.05 \cdot 10^{+77} \lor \neg \left(y.re \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{+48} \lor \neg \left(y.re \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.16e+48) (not (<= y.re 2.1e-45)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.16e+48) || !(y_46_re <= 2.1e-45)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.16d+48)) .or. (.not. (y_46re <= 2.1d-45))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.16e+48) || !(y_46_re <= 2.1e-45)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.16e+48) or not (y_46_re <= 2.1e-45):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.16e+48) || !(y_46_re <= 2.1e-45))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.16e+48) || ~((y_46_re <= 2.1e-45)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.16e+48], N[Not[LessEqual[y$46$re, 2.1e-45]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.15999999999999992e48 or 2.09999999999999995e-45 < y.re

   1. Initial program 49.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   if -1.15999999999999992e48 < y.re < 2.09999999999999995e-45

   1. Initial program 66.3%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 67.4

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

   5. Applied rewrites 67.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.16 \cdot 10^{+48} \lor \neg \left(y.re \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.1% 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 58.2%

   \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
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 40.5

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

5. Applied rewrites 40.5%

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

6. Final simplification 40.5%

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

Reproduce

herbie shell --seed 2024326 
(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))))