_divideComplex, imaginary part

Percentage Accurate: 61.8% → 84.8%

Time: 9.9s

Alternatives: 14

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, \frac{y.re}{t\_0} \cdot x.im\right)\\ \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \end{array} \end{array} \]

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.im -7.8e+121)
     (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
     (if (<= y.im -6.2e-61)
       t_1
       (if (<= y.im 2.1e-132)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 4.6e+141)
           t_1
           (fma (/ y.re y.im) (/ x.im y.im) (/ (- x.re) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_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_im <= -7.8e+121) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -6.2e-61) {
		tmp = t_1;
	} else if (y_46_im <= 2.1e-132) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.6e+141) {
		tmp = t_1;
	} else {
		tmp = fma((y_46_re / y_46_im), (x_46_im / y_46_im), (-x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_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_im <= -7.8e+121)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -6.2e-61)
		tmp = t_1;
	elseif (y_46_im <= 2.1e-132)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4.6e+141)
		tmp = t_1;
	else
		tmp = fma(Float64(y_46_re / y_46_im), Float64(x_46_im / y_46_im), Float64(Float64(-x_46_re) / y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$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$46$im, -7.8e+121], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -6.2e-61], t$95$1, If[LessEqual[y$46$im, 2.1e-132], 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, 4.6e+141], t$95$1, N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision] + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.79999999999999967e121

   1. Initial program 25.7%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 92.7%

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

   if -7.79999999999999967e121 < y.im < -6.1999999999999999e-61 or 2.1000000000000001e-132 < y.im < 4.6000000000000003e141

   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. 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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

   4. Applied rewrites 88.4%

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

   if -6.1999999999999999e-61 < y.im < 2.1000000000000001e-132

   1. Initial program 68.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 4.6000000000000003e141 < y.im

   1. Initial program 31.6%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   7. Applied rewrites 91.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.im}{y.im}, \frac{-x.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (- y.im) x.re (* x.im y.re)))
        (t_1 (fma y.im y.im (* y.re y.re))))
   (if (<= y.im -9.5e+45)
     (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
     (if (<= y.im -6.2e-61)
       (/ t_0 t_1)
       (if (<= y.im 4.4e-132)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 4.8e+30)
           (/ 1.0 (/ t_1 t_0))
           (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(-y_46_im, x_46_re, (x_46_im * y_46_re));
	double t_1 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -9.5e+45) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -6.2e-61) {
		tmp = t_0 / t_1;
	} else if (y_46_im <= 4.4e-132) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.8e+30) {
		tmp = 1.0 / (t_1 / t_0);
	} else {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re))
	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+45)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -6.2e-61)
		tmp = Float64(t_0 / t_1);
	elseif (y_46_im <= 4.4e-132)
		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+30)
		tmp = Float64(1.0 / Float64(t_1 / t_0));
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-y$46$im) * x$46$re + N[(x$46$im * y$46$re), $MachinePrecision]), $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+45], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -6.2e-61], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 4.4e-132], 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+30], N[(1.0 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.4999999999999998e45

   1. Initial program 41.6%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 89.6%

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

   if -9.4999999999999998e45 < y.im < -6.1999999999999999e-61

   1. Initial program 84.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 84.3

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

      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 84.3

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

   4. Applied rewrites 84.3%

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

   if -6.1999999999999999e-61 < y.im < 4.39999999999999981e-132

   1. Initial program 68.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 4.39999999999999981e-132 < y.im < 4.7999999999999999e30

   1. Initial program 90.5%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 90.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 90.6

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 4.7999999999999999e30 < y.im

   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.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 82.6%

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

Alternative 3: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.im -9.5e+45)
     (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
     (if (<= y.im -6.2e-61)
       (/ (fma (- y.im) x.re (* x.im y.re)) t_0)
       (if (<= y.im 4.4e-132)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 4.8e+30)
           (* (/ -1.0 t_0) (fma (- x.im) y.re (* x.re y.im)))
           (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -9.5e+45) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -6.2e-61) {
		tmp = fma(-y_46_im, x_46_re, (x_46_im * y_46_re)) / t_0;
	} else if (y_46_im <= 4.4e-132) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.8e+30) {
		tmp = (-1.0 / t_0) * fma(-x_46_im, y_46_re, (x_46_re * y_46_im));
	} else {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -9.5e+45)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -6.2e-61)
		tmp = Float64(fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re)) / t_0);
	elseif (y_46_im <= 4.4e-132)
		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+30)
		tmp = Float64(Float64(-1.0 / t_0) * fma(Float64(-x_46_im), y_46_re, Float64(x_46_re * y_46_im)));
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+45], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -6.2e-61], N[(N[((-y$46$im) * x$46$re + N[(x$46$im * y$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 4.4e-132], 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+30], 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], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.4999999999999998e45

   1. Initial program 41.6%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 89.6%

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

   if -9.4999999999999998e45 < y.im < -6.1999999999999999e-61

   1. Initial program 84.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 84.3

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

      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 84.3

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

   4. Applied rewrites 84.3%

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

   if -6.1999999999999999e-61 < y.im < 4.39999999999999981e-132

   1. Initial program 68.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 4.39999999999999981e-132 < y.im < 4.7999999999999999e30

   1. Initial program 90.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 90.5

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

      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 90.5

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

   4. Applied rewrites 90.5%

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

   if 4.7999999999999999e30 < y.im

   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.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 82.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (fma (- y.im) x.re (* x.im y.re)) (fma y.im y.im (* y.re y.re)))))
   (if (<= y.im -9.5e+45)
     (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
     (if (<= y.im -6.2e-61)
       t_0
       (if (<= y.im 4.4e-132)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 4.8e+30)
           t_0
           (/ (fma (/ y.re y.im) x.im (- x.re)) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(-y_46_im, x_46_re, (x_46_im * y_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -9.5e+45) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= -6.2e-61) {
		tmp = t_0;
	} else if (y_46_im <= 4.4e-132) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4.8e+30) {
		tmp = t_0;
	} else {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(Float64(-y_46_im), x_46_re, Float64(x_46_im * y_46_re)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_im <= -9.5e+45)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= -6.2e-61)
		tmp = t_0;
	elseif (y_46_im <= 4.4e-132)
		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+30)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[((-y$46$im) * x$46$re + 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]}, If[LessEqual[y$46$im, -9.5e+45], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -6.2e-61], t$95$0, If[LessEqual[y$46$im, 4.4e-132], 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+30], t$95$0, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.4999999999999998e45

   1. Initial program 41.6%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 89.6%

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

   if -9.4999999999999998e45 < y.im < -6.1999999999999999e-61 or 4.39999999999999981e-132 < y.im < 4.7999999999999999e30

   1. Initial program 88.1%

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

      1. lift--.f64 N/A

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

      2. 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 88.1

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

      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 88.1

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

   4. Applied rewrites 88.1%

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

   if -6.1999999999999999e-61 < y.im < 4.39999999999999981e-132

   1. Initial program 68.5%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if 4.7999999999999999e30 < y.im

   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.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 82.6%

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

Alternative 5: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -3.7 \cdot 10^{-200}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -4.5e-18)
     t_0
     (if (<= y.im -3.7e-200)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.re y.re))
       (if (<= y.im 8e-106)
         (/ x.im y.re)
         (if (<= y.im 3.2e+115)
           (* (/ y.im (fma y.im y.im (* y.re y.re))) (- x.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 <= -4.5e-18) {
		tmp = t_0;
	} else if (y_46_im <= -3.7e-200) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_re * y_46_re);
	} else if (y_46_im <= 8e-106) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.2e+115) {
		tmp = (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * -x_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 <= -4.5e-18)
		tmp = t_0;
	elseif (y_46_im <= -3.7e-200)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 8e-106)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.2e+115)
		tmp = Float64(Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * Float64(-x_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, -4.5e-18], t$95$0, If[LessEqual[y$46$im, -3.7e-200], 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], If[LessEqual[y$46$im, 8e-106], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.2e+115], N[(N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.49999999999999994e-18 or 3.2e115 < y.im

   1. Initial program 45.4%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if -4.49999999999999994e-18 < y.im < -3.70000000000000011e-200

   1. Initial program 80.5%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{{y.re}^{2}}} \]
   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 64.5

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

   5. Applied rewrites 64.5%

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

   if -3.70000000000000011e-200 < y.im < 7.99999999999999953e-106

   1. Initial program 64.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if 7.99999999999999953e-106 < y.im < 3.2e115

   1. Initial program 80.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 64.8

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

   8. Applied rewrites 64.8%

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

Alternative 6: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1 (* (/ y.im (fma y.im y.im (* y.re y.re))) (- x.re))))
   (if (<= y.im -7.4e+121)
     t_0
     (if (<= y.im -7.5e-66)
       t_1
       (if (<= y.im 8e-106) (/ x.im y.re) (if (<= y.im 3.2e+115) t_1 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_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * -x_46_re;
	double tmp;
	if (y_46_im <= -7.4e+121) {
		tmp = t_0;
	} else if (y_46_im <= -7.5e-66) {
		tmp = t_1;
	} else if (y_46_im <= 8e-106) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.2e+115) {
		tmp = t_1;
	} 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_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * Float64(-x_46_re))
	tmp = 0.0
	if (y_46_im <= -7.4e+121)
		tmp = t_0;
	elseif (y_46_im <= -7.5e-66)
		tmp = t_1;
	elseif (y_46_im <= 8e-106)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.2e+115)
		tmp = t_1;
	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$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision]}, If[LessEqual[y$46$im, -7.4e+121], t$95$0, If[LessEqual[y$46$im, -7.5e-66], t$95$1, If[LessEqual[y$46$im, 8e-106], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.2e+115], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.40000000000000025e121 or 3.2e115 < y.im

   1. Initial program 30.0%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -7.40000000000000025e121 < y.im < -7.49999999999999995e-66 or 7.99999999999999953e-106 < y.im < 3.2e115

   1. Initial program 80.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 32.0

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

   5. Applied rewrites 32.0%

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

   6. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if -7.49999999999999995e-66 < y.im < 7.99999999999999953e-106

   1. Initial program 69.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

Alternative 7: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.15e+46)
     t_0
     (if (<= y.im 8.5e-55)
       (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
       (if (<= y.im 3.2e+115)
         (* (/ y.im (fma y.im y.im (* y.re y.re))) (- x.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 <= -2.15e+46) {
		tmp = t_0;
	} else if (y_46_im <= 8.5e-55) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 3.2e+115) {
		tmp = (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * -x_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 <= -2.15e+46)
		tmp = t_0;
	elseif (y_46_im <= 8.5e-55)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 3.2e+115)
		tmp = Float64(Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * Float64(-x_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, -2.15e+46], t$95$0, If[LessEqual[y$46$im, 8.5e-55], 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, 3.2e+115], N[(N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.15000000000000002e46 or 3.2e115 < y.im

   1. Initial program 39.3%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -2.15000000000000002e46 < y.im < 8.49999999999999968e-55

   1. Initial program 72.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 8.49999999999999968e-55 < y.im < 3.2e115

   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.im around 0

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

      1. lower-/.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in x.im around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 68.5

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

   8. Applied rewrites 68.5%

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

Alternative 8: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+170}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.75e+28)
   (/ x.im y.re)
   (if (<= y.re 3.2e-104)
     (/ (- x.re) y.im)
     (if (<= y.re 4e+170)
       (* (/ y.re (fma y.im y.im (* y.re y.re))) 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 <= -1.75e+28) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 3.2e-104) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4e+170) {
		tmp = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * 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 <= -1.75e+28)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 3.2e-104)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 4e+170)
		tmp = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * 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, -1.75e+28], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.2e-104], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4e+170], N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.75e28 or 4.00000000000000014e170 < y.re

   1. Initial program 48.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -1.75e28 < y.re < 3.19999999999999989e-104

   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.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.6

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

   5. Applied rewrites 68.6%

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

   if 3.19999999999999989e-104 < y.re < 4.00000000000000014e170

   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.im around 0

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

      1. lower-/.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in x.im around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.3

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

   8. Applied rewrites 50.3%

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

Alternative 9: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, \frac{x.re}{y.re}, -x.im\right)}{-y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.im (/ x.re y.re) (- x.im)) (- y.re))))
   (if (<= y.re -3.5e+27)
     t_0
     (if (<= y.re 2.7e-22) (/ (fma (/ y.re y.im) x.im (- x.re)) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, (x_46_re / y_46_re), -x_46_im) / -y_46_re;
	double tmp;
	if (y_46_re <= -3.5e+27) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-22) {
		tmp = fma((y_46_re / y_46_im), x_46_im, -x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_im, Float64(x_46_re / y_46_re), Float64(-x_46_im)) / Float64(-y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.5e+27)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-22)
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision] + (-x$46$im)), $MachinePrecision] / (-y$46$re)), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+27], t$95$0, If[LessEqual[y$46$re, 2.7e-22], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.5000000000000002e27 or 2.7000000000000002e-22 < y.re

   1. Initial program 55.2%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in y.re around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot x.im + \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   7. 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 74.6

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

   8. Applied rewrites 74.6%

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

   if -3.5000000000000002e27 < y.re < 2.7000000000000002e-22

   1. Initial program 68.4%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

   7. Applied rewrites 84.7%

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

Alternative 10: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.15e+46)
   (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)
   (if (<= y.im 8.8e-47)
     (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
     (/ (fma (/ y.re y.im) x.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_im <= -2.15e+46) {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_im <= 8.8e-47) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = fma((y_46_re / y_46_im), x_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_im <= -2.15e+46)
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_im <= 8.8e-47)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(-x_46_re)) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.15e+46], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 8.8e-47], 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 / y$46$im), $MachinePrecision] * x$46$im + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.15000000000000002e46

   1. Initial program 42.4%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   7. Applied rewrites 91.2%

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

   if -2.15000000000000002e46 < y.im < 8.80000000000000075e-47

   1. Initial program 73.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if 8.80000000000000075e-47 < y.im

   1. Initial program 57.0%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   7. Applied rewrites 77.5%

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

Alternative 11: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.15 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) y.im)))
   (if (<= y.im -2.15e+46)
     t_0
     (if (<= y.im 8.8e-47) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.15e+46) {
		tmp = t_0;
	} else if (y_46_im <= 8.8e-47) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.15e+46)
		tmp = t_0;
	elseif (y_46_im <= 8.8e-47)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.15e+46], t$95$0, If[LessEqual[y$46$im, 8.8e-47], 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 < -2.15000000000000002e46 or 8.80000000000000075e-47 < y.im

   1. Initial program 50.9%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 82.4%

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

   if -2.15000000000000002e46 < y.im < 8.80000000000000075e-47

   1. Initial program 73.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

Alternative 12: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (/ (* x.re y.im) y.re)) y.re)))
   (if (<= y.re -3.3e-20)
     t_0
     (if (<= y.re 2.7e-22) (/ (- (/ (* x.im y.re) y.im) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -3.3e-20) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-22) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} 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_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    if (y_46re <= (-3.3d-20)) then
        tmp = t_0
    else if (y_46re <= 2.7d-22) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    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_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -3.3e-20) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-22) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} 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_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -3.3e-20:
		tmp = t_0
	elif y_46_re <= 2.7e-22:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.3e-20)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-22)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.3e-20)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-22)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	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[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.3e-20], t$95$0, If[LessEqual[y$46$re, 2.7e-22], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.3e-20 or 2.7000000000000002e-22 < y.re

   1. Initial program 55.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -3.3e-20 < y.re < 2.7000000000000002e-22

   1. Initial program 68.7%

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

   3. Taylor expanded in y.im around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

Alternative 13: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.52 \cdot 10^{+93}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.75e+28)
   (/ x.im y.re)
   (if (<= y.re 1.52e+93) (/ (- 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 <= -1.75e+28) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.52e+93) {
		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 <= (-1.75d+28)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 1.52d+93) 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 <= -1.75e+28) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 1.52e+93) {
		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 <= -1.75e+28:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 1.52e+93:
		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 <= -1.75e+28)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 1.52e+93)
		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 <= -1.75e+28)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 1.52e+93)
		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, -1.75e+28], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.52e+93], 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 < -1.75e28 or 1.52e93 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.75e28 < y.re < 1.52e93

   1. Initial program 67.9%

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

   3. Taylor expanded in y.im around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.5

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

   5. Applied rewrites 62.5%

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

Alternative 14: 43.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in y.im around 0

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

   1. lower-/.f64 35.8

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

5. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2024277 
(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))))