_divideComplex, imaginary part

Percentage Accurate: 61.4% → 92.1%

Time: 10.2s

Alternatives: 13

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, t\_0\right)\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{y.im}, t\_0\right)\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, \mathsf{fma}\left(y.re, \frac{y.re}{y.im \cdot y.im}, -1\right) \cdot x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (fma y.im (/ y.im y.re) y.re)))
        (t_1 (fma (- x.re) (/ y.im (fma y.im y.im (* y.re y.re))) t_0)))
   (if (<= y.im -4e+154)
     (fma (- x.re) (/ 1.0 y.im) t_0)
     (if (<= y.im -2.9e-143)
       t_1
       (if (<= y.im 5e-179)
         (/ (- x.im (/ (* x.re y.im) y.re)) y.re)
         (if (<= y.im 9e+150)
           t_1
           (/
            (fma
             (/ y.re y.im)
             x.im
             (* (fma y.re (/ y.re (* y.im y.im)) -1.0) 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 = x_46_im / fma(y_46_im, (y_46_im / y_46_re), y_46_re);
	double t_1 = fma(-x_46_re, (y_46_im / fma(y_46_im, y_46_im, (y_46_re * y_46_re))), t_0);
	double tmp;
	if (y_46_im <= -4e+154) {
		tmp = fma(-x_46_re, (1.0 / y_46_im), t_0);
	} else if (y_46_im <= -2.9e-143) {
		tmp = t_1;
	} else if (y_46_im <= 5e-179) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 9e+150) {
		tmp = t_1;
	} else {
		tmp = fma((y_46_re / y_46_im), x_46_im, (fma(y_46_re, (y_46_re / (y_46_im * y_46_im)), -1.0) * 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(x_46_im / fma(y_46_im, Float64(y_46_im / y_46_re), y_46_re))
	t_1 = fma(Float64(-x_46_re), Float64(y_46_im / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))), t_0)
	tmp = 0.0
	if (y_46_im <= -4e+154)
		tmp = fma(Float64(-x_46_re), Float64(1.0 / y_46_im), t_0);
	elseif (y_46_im <= -2.9e-143)
		tmp = t_1;
	elseif (y_46_im <= 5e-179)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 9e+150)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_im, Float64(fma(y_46_re, Float64(y_46_re / Float64(y_46_im * y_46_im)), -1.0) * 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[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x$46$re) * N[(y$46$im / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+154], N[((-x$46$re) * N[(1.0 / y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[y$46$im, -2.9e-143], t$95$1, If[LessEqual[y$46$im, 5e-179], 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, 9e+150], t$95$1, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im + N[(N[(y$46$re * N[(y$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.00000000000000015e154

   1. Initial program 30.1%

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

      1. lift-/.f64 N/A

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

      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 31.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 31.3

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

   6. Applied rewrites 31.3%

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

   7. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 45.6

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

   9. Applied rewrites 45.6%

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

   10. Taylor expanded in y.im around inf

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

       1. lower-/.f64 91.7

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

   12. Applied rewrites 91.7%

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

   if -4.00000000000000015e154 < y.im < -2.9000000000000001e-143 or 4.9999999999999998e-179 < y.im < 9.00000000000000001e150

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-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 79.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 80.0

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

   6. Applied rewrites 80.0%

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

   7. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 96.3

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

   9. Applied rewrites 96.3%

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

   if -2.9000000000000001e-143 < y.im < 4.9999999999999998e-179

   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 94.7

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

   5. Applied rewrites 94.7%

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

   if 9.00000000000000001e150 < y.im

   1. Initial program 38.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 10.6

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

   5. Applied rewrites 10.6%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

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

   9. Taylor expanded in y.im around 0

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

   11. Applied rewrites 10.5%

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

   12. Taylor expanded in y.im around inf

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

       1. lower-/.f64 N/A

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

   14. Applied rewrites 92.8%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{y.im}, \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\right)\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\right)\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.im, \mathsf{fma}\left(y.re, \frac{y.re}{y.im \cdot y.im}, -1\right) \cdot x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-x.re, \frac{y.im}{t\_0}, \frac{y.re}{t\_0} \cdot x.im\right)\\ t_2 := \mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -1.38 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-146}:\\ \;\;\;\;\mathsf{fma}\left(-x.re, \frac{1}{y.im}, \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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)))
        (t_2 (fma (/ (- x.re) y.re) (/ y.im y.re) (/ x.im y.re))))
   (if (<= y.re -7e+140)
     t_2
     (if (<= y.re -1.38e-157)
       t_1
       (if (<= y.re 6e-146)
         (fma (- x.re) (/ 1.0 y.im) (/ x.im (fma y.im (/ y.im y.re) y.re)))
         (if (<= y.re 6e+75) t_1 t_2))))))

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 t_2 = fma((-x_46_re / y_46_re), (y_46_im / y_46_re), (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -7e+140) {
		tmp = t_2;
	} else if (y_46_re <= -1.38e-157) {
		tmp = t_1;
	} else if (y_46_re <= 6e-146) {
		tmp = fma(-x_46_re, (1.0 / y_46_im), (x_46_im / fma(y_46_im, (y_46_im / y_46_re), y_46_re)));
	} else if (y_46_re <= 6e+75) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(Float64(Float64(-x_46_re) / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -7e+140)
		tmp = t_2;
	elseif (y_46_re <= -1.38e-157)
		tmp = t_1;
	elseif (y_46_re <= 6e-146)
		tmp = fma(Float64(-x_46_re), Float64(1.0 / y_46_im), Float64(x_46_im / fma(y_46_im, Float64(y_46_im / y_46_re), y_46_re)));
	elseif (y_46_re <= 6e+75)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[((-x$46$re) / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7e+140], t$95$2, If[LessEqual[y$46$re, -1.38e-157], t$95$1, If[LessEqual[y$46$re, 6e-146], N[((-x$46$re) * N[(1.0 / y$46$im), $MachinePrecision] + N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6e+75], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.99999999999999978e140 or 6e75 < y.re

   1. Initial program 36.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in y.im around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.0

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

   8. Applied rewrites 92.0%

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

   if -6.99999999999999978e140 < y.re < -1.3799999999999999e-157 or 6.00000000000000038e-146 < y.re < 6e75

   1. Initial program 74.1%

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

      1. lift-/.f64 N/A

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

      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 81.6%

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

   if -1.3799999999999999e-157 < y.re < 6.00000000000000038e-146

   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. 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 64.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 64.2

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

   6. Applied rewrites 64.2%

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

   7. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in y.im around inf

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

       1. lower-/.f64 93.9

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

   12. Applied rewrites 93.9%

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

Alternative 3: 66.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{t\_1}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)) (t_1 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.im -2.05e+58)
     t_0
     (if (<= y.im -3.5e-60)
       (/ t_1 (* y.im y.im))
       (if (<= y.im -2.6e-148)
         (/ t_1 (* y.re y.re))
         (if (<= y.im 8.5e-115)
           (/ x.im y.re)
           (if (<= y.im 3.1e+124)
             (* (/ 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 t_1 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_im <= -2.05e+58) {
		tmp = t_0;
	} else if (y_46_im <= -3.5e-60) {
		tmp = t_1 / (y_46_im * y_46_im);
	} else if (y_46_im <= -2.6e-148) {
		tmp = t_1 / (y_46_re * y_46_re);
	} else if (y_46_im <= 8.5e-115) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.1e+124) {
		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)
	t_1 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.05e+58)
		tmp = t_0;
	elseif (y_46_im <= -3.5e-60)
		tmp = Float64(t_1 / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= -2.6e-148)
		tmp = Float64(t_1 / Float64(y_46_re * y_46_re));
	elseif (y_46_im <= 8.5e-115)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.1e+124)
		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]}, Block[{t$95$1 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.05e+58], t$95$0, If[LessEqual[y$46$im, -3.5e-60], N[(t$95$1 / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.6e-148], N[(t$95$1 / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.5e-115], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+124], 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 5 regimes

2. if y.im < -2.05e58 or 3.1000000000000002e124 < y.im

   1. Initial program 40.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -2.05e58 < y.im < -3.49999999999999976e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -3.49999999999999976e-60 < y.im < -2.60000000000000008e-148

   1. Initial program 91.1%

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

   3. Taylor expanded in y.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 79.1

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

   5. Applied rewrites 79.1%

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

   if -2.60000000000000008e-148 < y.im < 8.49999999999999953e-115

   1. Initial program 74.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 8.49999999999999953e-115 < y.im < 3.1000000000000002e124

   1. Initial program 62.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 35.9

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

   5. Applied rewrites 35.9%

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

   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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+58}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-x.re}{y.re}, \frac{y.im}{y.re}, \frac{x.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)}{-\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma (/ (- x.re) y.re) (/ y.im y.re) (/ x.im y.re))))
   (if (<= y.re -2.9e-34)
     t_0
     (if (<= y.re 9e-146)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (if (<= y.re 6.5e+137)
         (/
          (fma (- x.im) y.re (* x.re y.im))
          (- (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma((-x_46_re / y_46_re), (y_46_im / y_46_re), (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -2.9e-34) {
		tmp = t_0;
	} else if (y_46_re <= 9e-146) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.5e+137) {
		tmp = fma(-x_46_im, y_46_re, (x_46_re * y_46_im)) / -fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(Float64(Float64(-x_46_re) / y_46_re), Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.9e-34)
		tmp = t_0;
	elseif (y_46_re <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6.5e+137)
		tmp = Float64(fma(Float64(-x_46_im), y_46_re, Float64(x_46_re * y_46_im)) / Float64(-fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[((-x$46$re) / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision] + N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e-34], t$95$0, If[LessEqual[y$46$re, 9e-146], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.5e+137], N[(N[((-x$46$im) * y$46$re + N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / (-N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9000000000000002e-34 or 6.5000000000000002e137 < y.re

   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 0

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y.im around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 82.7

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

   8. Applied rewrites 82.7%

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

   if -2.9000000000000002e-34 < y.re < 9.0000000000000001e-146

   1. Initial program 70.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   if 9.0000000000000001e-146 < y.re < 6.5000000000000002e137

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 78.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 5: 81.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-x.re, \frac{1}{y.im}, \frac{x.im}{\mathsf{fma}\left(y.im, \frac{y.im}{y.re}, y.re\right)}\right)\\ \mathbf{if}\;y.im \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma (- x.re) (/ 1.0 y.im) (/ x.im (fma y.im (/ y.im y.re) y.re)))))
   (if (<= y.im -3.5e-60)
     t_0
     (if (<= y.im 5.2e-31) (/ (- 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(-x_46_re, (1.0 / y_46_im), (x_46_im / fma(y_46_im, (y_46_im / y_46_re), y_46_re)));
	double tmp;
	if (y_46_im <= -3.5e-60) {
		tmp = t_0;
	} else if (y_46_im <= 5.2e-31) {
		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 = fma(Float64(-x_46_re), Float64(1.0 / y_46_im), Float64(x_46_im / fma(y_46_im, Float64(y_46_im / y_46_re), y_46_re)))
	tmp = 0.0
	if (y_46_im <= -3.5e-60)
		tmp = t_0;
	elseif (y_46_im <= 5.2e-31)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) * N[(1.0 / y$46$im), $MachinePrecision] + N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision] + y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.5e-60], t$95$0, If[LessEqual[y$46$im, 5.2e-31], 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 < -3.49999999999999976e-60 or 5.19999999999999991e-31 < y.im

   1. Initial program 51.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. 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 59.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 59.5

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

   6. Applied rewrites 59.5%

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

   7. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 74.7

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

   9. Applied rewrites 74.7%

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

   10. Taylor expanded in y.im around inf

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

       1. lower-/.f64 80.8

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

   12. Applied rewrites 80.8%

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

   if -3.49999999999999976e-60 < y.im < 5.19999999999999991e-31

   1. Initial program 77.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

Alternative 6: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x.im, y.re, x.re \cdot y.im\right)}{-\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.9e-34)
     t_0
     (if (<= y.re 9e-146)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (if (<= y.re 2.8e+139)
         (/
          (fma (- x.im) y.re (* x.re y.im))
          (- (fma y.im y.im (* y.re y.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.9e-34) {
		tmp = t_0;
	} else if (y_46_re <= 9e-146) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.8e+139) {
		tmp = fma(-x_46_im, y_46_re, (x_46_re * y_46_im)) / -fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		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 <= -2.9e-34)
		tmp = t_0;
	elseif (y_46_re <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.8e+139)
		tmp = Float64(fma(Float64(-x_46_im), y_46_re, Float64(x_46_re * y_46_im)) / Float64(-fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))));
	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[(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, -2.9e-34], t$95$0, If[LessEqual[y$46$re, 9e-146], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.8e+139], N[(N[((-x$46$im) * y$46$re + N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / (-N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9000000000000002e-34 or 2.7999999999999998e139 < y.re

   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 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 76.4

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

   5. Applied rewrites 76.4%

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

   if -2.9000000000000002e-34 < y.re < 9.0000000000000001e-146

   1. Initial program 70.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   if 9.0000000000000001e-146 < y.re < 2.7999999999999998e139

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 78.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 7: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2e+138)
     t_0
     (if (<= y.im -1.25e-96)
       t_1
       (if (<= y.im 8.5e-115)
         (/ x.im y.re)
         (if (<= y.im 3.1e+124) 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 <= -2e+138) {
		tmp = t_0;
	} else if (y_46_im <= -1.25e-96) {
		tmp = t_1;
	} else if (y_46_im <= 8.5e-115) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 3.1e+124) {
		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 <= -2e+138)
		tmp = t_0;
	elseif (y_46_im <= -1.25e-96)
		tmp = t_1;
	elseif (y_46_im <= 8.5e-115)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 3.1e+124)
		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, -2e+138], t$95$0, If[LessEqual[y$46$im, -1.25e-96], t$95$1, If[LessEqual[y$46$im, 8.5e-115], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+124], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.0000000000000001e138 or 3.1000000000000002e124 < y.im

   1. Initial program 37.0%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   if -2.0000000000000001e138 < y.im < -1.24999999999999999e-96 or 8.49999999999999953e-115 < y.im < 3.1000000000000002e124

   1. Initial program 66.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 36.0

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

   5. Applied rewrites 36.0%

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

   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. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 56.7

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

   8. Applied rewrites 56.7%

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

   if -1.24999999999999999e-96 < y.im < 8.49999999999999953e-115

   1. Initial program 77.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-96}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \left(-x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.9e-34)
     t_0
     (if (<= y.re 9e-146)
       (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
       (if (<= y.re 2.8e+139)
         (/ (fma (- y.im) x.re (* x.im y.re)) (fma y.im y.im (* y.re y.re)))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -2.9e-34) {
		tmp = t_0;
	} else if (y_46_re <= 9e-146) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.8e+139) {
		tmp = 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));
	} 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 <= -2.9e-34)
		tmp = t_0;
	elseif (y_46_re <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.8e+139)
		tmp = 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)));
	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[(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, -2.9e-34], t$95$0, If[LessEqual[y$46$re, 9e-146], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.8e+139], 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], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9000000000000002e-34 or 2.7999999999999998e139 < y.re

   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 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 76.4

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

   5. Applied rewrites 76.4%

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

   if -2.9000000000000002e-34 < y.re < 9.0000000000000001e-146

   1. Initial program 70.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.9

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

   5. Applied rewrites 87.9%

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

   if 9.0000000000000001e-146 < y.re < 2.7999999999999998e139

   1. Initial program 78.0%

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

      1. lift--.f64 N/A

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

      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 78.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

Alternative 9: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.48 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ y.re (fma y.im y.im (* y.re y.re))) x.im)))
   (if (<= y.re -1.9e+143)
     (/ x.im y.re)
     (if (<= y.re -3.8e-35)
       t_0
       (if (<= y.re 7.2e-116)
         (/ (- x.re) y.im)
         (if (<= y.re 1.48e+113) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re / fma(y_46_im, y_46_im, (y_46_re * y_46_re))) * x_46_im;
	double tmp;
	if (y_46_re <= -1.9e+143) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.8e-35) {
		tmp = t_0;
	} else if (y_46_re <= 7.2e-116) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 1.48e+113) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))) * x_46_im)
	tmp = 0.0
	if (y_46_re <= -1.9e+143)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.8e-35)
		tmp = t_0;
	elseif (y_46_re <= 7.2e-116)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 1.48e+113)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y$46$re / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -1.9e+143], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.8e-35], t$95$0, If[LessEqual[y$46$re, 7.2e-116], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.48e+113], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.9e143 or 1.48000000000000002e113 < y.re

   1. Initial program 33.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.7

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

   5. Applied rewrites 81.7%

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

   if -1.9e143 < y.re < -3.8000000000000001e-35 or 7.19999999999999951e-116 < y.re < 1.48000000000000002e113

   1. Initial program 73.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-/.f64 36.1

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

   5. Applied rewrites 36.1%

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

   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.4

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

   8. Applied rewrites 55.4%

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

   if -3.8000000000000001e-35 < y.re < 7.19999999999999951e-116

   1. Initial program 71.8%

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

   3. Taylor expanded in y.im around 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 67.4

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

   5. Applied rewrites 67.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.48 \cdot 10^{+113}:\\ \;\;\;\;\frac{y.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -2.05e+58)
     t_0
     (if (<= y.im -7.8e-60)
       (/ (- (* x.im y.re) (* x.re y.im)) (* y.im y.im))
       (if (<= y.im 2.1e+122) (/ (- x.im (/ (* x.re y.im) y.re)) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.05e+58) {
		tmp = t_0;
	} else if (y_46_im <= -7.8e-60) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.1e+122) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-2.05d+58)) then
        tmp = t_0
    else if (y_46im <= (-7.8d-60)) then
        tmp = ((x_46im * y_46re) - (x_46re * y_46im)) / (y_46im * y_46im)
    else if (y_46im <= 2.1d+122) then
        tmp = (x_46im - ((x_46re * y_46im) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -2.05e+58) {
		tmp = t_0;
	} else if (y_46_im <= -7.8e-60) {
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	} else if (y_46_im <= 2.1e+122) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -2.05e+58:
		tmp = t_0
	elif y_46_im <= -7.8e-60:
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im)
	elif y_46_im <= 2.1e+122:
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.05e+58)
		tmp = t_0;
	elseif (y_46_im <= -7.8e-60)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(y_46_im * y_46_im));
	elseif (y_46_im <= 2.1e+122)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.05e+58)
		tmp = t_0;
	elseif (y_46_im <= -7.8e-60)
		tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / (y_46_im * y_46_im);
	elseif (y_46_im <= 2.1e+122)
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.05e+58], t$95$0, If[LessEqual[y$46$im, -7.8e-60], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.1e+122], N[(N[(x$46$im - N[(N[(x$46$re * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.05e58 or 2.10000000000000016e122 < y.im

   1. Initial program 40.5%

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

   3. Taylor expanded in y.im around 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 75.5

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

   5. Applied rewrites 75.5%

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

   if -2.05e58 < y.im < -7.8000000000000004e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -7.8000000000000004e-60 < y.im < 2.10000000000000016e122

   1. Initial program 72.9%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

Alternative 11: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2.9e-34)
     t_0
     (if (<= y.re 1.02e-51) (/ (- (/ (* 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 <= -2.9e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-51) {
		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 <= (-2.9d-34)) then
        tmp = t_0
    else if (y_46re <= 1.02d-51) 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 <= -2.9e-34) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-51) {
		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 <= -2.9e-34:
		tmp = t_0
	elif y_46_re <= 1.02e-51:
		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 <= -2.9e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-51)
		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 <= -2.9e-34)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-51)
		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, -2.9e-34], t$95$0, If[LessEqual[y$46$re, 1.02e-51], 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 < -2.9000000000000002e-34 or 1.01999999999999998e-51 < 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

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   if -2.9000000000000002e-34 < y.re < 1.01999999999999998e-51

   1. Initial program 73.2%

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

   3. Taylor expanded in y.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 83.9

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

   5. Applied rewrites 83.9%

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

Alternative 12: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -6.8e-40) t_0 (if (<= y.im 3.8e-30) (/ x.im y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -6.8e-40) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-30) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-6.8d-40)) then
        tmp = t_0
    else if (y_46im <= 3.8d-30) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -6.8e-40) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-30) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -6.8e-40:
		tmp = t_0
	elif y_46_im <= 3.8e-30:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.8e-40)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-30)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -6.8e-40)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-30)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6.8e-40], t$95$0, If[LessEqual[y$46$im, 3.8e-30], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.79999999999999968e-40 or 3.8000000000000003e-30 < y.im

   1. Initial program 51.4%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 62.0

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

   5. Applied rewrites 62.0%

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

   if -6.79999999999999968e-40 < y.im < 3.8000000000000003e-30

   1. Initial program 75.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 63.9

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

   5. Applied rewrites 63.9%

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

Alternative 13: 42.9% 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 42.4

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

5. Applied rewrites 42.4%

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

Reproduce

herbie shell --seed 2024249 
(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))))