_divideComplex, imaginary part

Percentage Accurate: 61.6% → 83.1%

Time: 9.9s

Alternatives: 8

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.72 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{t\_0}, y.re, \frac{-x.re \cdot y.im}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \frac{x.re}{t\_0}, \frac{y.re \cdot x.im}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (/ (fma y.im (/ x.re (- y.re)) x.im) y.re)))
   (if (<= y.re -1.72e+84)
     t_1
     (if (<= y.re -7.5e-136)
       (fma (/ x.im t_0) y.re (/ (- (* x.re y.im)) t_0))
       (if (<= y.re 1.95e-14)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 6.6e+92)
           (fma (- y.im) (/ x.re t_0) (/ (* y.re x.im) t_0))
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(y_46_im, (x_46_re / -y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -1.72e+84) {
		tmp = t_1;
	} else if (y_46_re <= -7.5e-136) {
		tmp = fma((x_46_im / t_0), y_46_re, (-(x_46_re * y_46_im) / t_0));
	} else if (y_46_re <= 1.95e-14) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 6.6e+92) {
		tmp = fma(-y_46_im, (x_46_re / t_0), ((y_46_re * x_46_im) / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = Float64(fma(y_46_im, Float64(x_46_re / Float64(-y_46_re)), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.72e+84)
		tmp = t_1;
	elseif (y_46_re <= -7.5e-136)
		tmp = fma(Float64(x_46_im / t_0), y_46_re, Float64(Float64(-Float64(x_46_re * y_46_im)) / t_0));
	elseif (y_46_re <= 1.95e-14)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 6.6e+92)
		tmp = fma(Float64(-y_46_im), Float64(x_46_re / t_0), Float64(Float64(y_46_re * x_46_im) / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$im * N[(x$46$re / (-y$46$re)), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.72e+84], t$95$1, If[LessEqual[y$46$re, -7.5e-136], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$re + N[((-N[(x$46$re * y$46$im), $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.95e-14], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+92], N[((-y$46$im) * N[(x$46$re / t$95$0), $MachinePrecision] + N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.72e84 or 6.59999999999999948e92 < y.re

   1. Initial program 50.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 50.0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -1.72e84 < y.re < -7.5000000000000003e-136

   1. Initial program 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. distribute-neg-frac2 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 91.5%

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

   if -7.5000000000000003e-136 < y.re < 1.9499999999999999e-14

   1. Initial program 69.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 92.9%

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

   if 1.9499999999999999e-14 < y.re < 6.59999999999999948e92

   1. Initial program 68.8%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 75.6

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

   4. Applied rewrites 75.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.72 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, y.re, \frac{-x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(-y.im, \frac{x.re}{t\_0}, \frac{y.re \cdot x.im}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -3 \cdot 10^{+22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (- y.im) (/ x.re t_0) (/ (* y.re x.im) t_0)))
        (t_2 (/ (fma y.im (/ x.re (- y.re)) x.im) y.re)))
   (if (<= y.re -3e+22)
     t_2
     (if (<= y.re -4e-133)
       t_1
       (if (<= y.re 1.95e-14)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 6.6e+92) 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(-y_46_im, (x_46_re / t_0), ((y_46_re * x_46_im) / t_0));
	double t_2 = fma(y_46_im, (x_46_re / -y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -3e+22) {
		tmp = t_2;
	} else if (y_46_re <= -4e-133) {
		tmp = t_1;
	} else if (y_46_re <= 1.95e-14) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 6.6e+92) {
		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(-y_46_im), Float64(x_46_re / t_0), Float64(Float64(y_46_re * x_46_im) / t_0))
	t_2 = Float64(fma(y_46_im, Float64(x_46_re / Float64(-y_46_re)), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3e+22)
		tmp = t_2;
	elseif (y_46_re <= -4e-133)
		tmp = t_1;
	elseif (y_46_re <= 1.95e-14)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 6.6e+92)
		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[((-y$46$im) * N[(x$46$re / t$95$0), $MachinePrecision] + N[(N[(y$46$re * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$im * N[(x$46$re / (-y$46$re)), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3e+22], t$95$2, If[LessEqual[y$46$re, -4e-133], t$95$1, If[LessEqual[y$46$re, 1.95e-14], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+92], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3e22 or 6.59999999999999948e92 < y.re

   1. Initial program 52.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if -3e22 < y.re < -4.0000000000000003e-133 or 1.9499999999999999e-14 < y.re < 6.59999999999999948e92

   1. Initial program 76.8%

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

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 82.5

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

   4. Applied rewrites 82.5%

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

   if -4.0000000000000003e-133 < y.re < 1.9499999999999999e-14

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 90.2

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

   5. Applied rewrites 90.2%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-y.im, \frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}, \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ t_1 := \frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* y.re x.im) (* x.re y.im)) (fma y.re y.re (* y.im y.im))))
        (t_1 (/ (fma y.im (/ x.re (- y.re)) x.im) y.re)))
   (if (<= y.re -1.32e+22)
     t_1
     (if (<= y.re -1.6e-123)
       t_0
       (if (<= y.re 1.95e-14)
         (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)
         (if (<= y.re 6.4e+69) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	double t_1 = fma(y_46_im, (x_46_re / -y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -1.32e+22) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-123) {
		tmp = t_0;
	} else if (y_46_re <= 1.95e-14) {
		tmp = fma(x_46_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	} else if (y_46_re <= 6.4e+69) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)))
	t_1 = Float64(fma(y_46_im, Float64(x_46_re / Float64(-y_46_re)), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.32e+22)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-123)
		tmp = t_0;
	elseif (y_46_re <= 1.95e-14)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	elseif (y_46_re <= 6.4e+69)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$im * N[(x$46$re / (-y$46$re)), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.32e+22], t$95$1, If[LessEqual[y$46$re, -1.6e-123], t$95$0, If[LessEqual[y$46$re, 1.95e-14], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.4e+69], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.32e22 or 6.3999999999999997e69 < y.re

   1. Initial program 52.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 52.1

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 82.7

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

   7. Applied rewrites 82.7%

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

   if -1.32e22 < y.re < -1.59999999999999989e-123 or 1.9499999999999999e-14 < y.re < 6.3999999999999997e69

   1. Initial program 80.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 80.4

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

   4. Applied rewrites 80.4%

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

   if -1.59999999999999989e-123 < y.re < 1.9499999999999999e-14

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 91.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{if}\;y.re \leq -1700000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.im (/ x.re (- y.re)) x.im) y.re)))
   (if (<= y.re -1700000000000.0)
     t_0
     (if (<= y.re 5e+69) (/ (fma 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 = fma(y_46_im, (x_46_re / -y_46_re), x_46_im) / y_46_re;
	double tmp;
	if (y_46_re <= -1700000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 5e+69) {
		tmp = fma(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(fma(y_46_im, Float64(x_46_re / Float64(-y_46_re)), x_46_im) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1700000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 5e+69)
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * N[(x$46$re / (-y$46$re)), $MachinePrecision] + x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1700000000000.0], t$95$0, If[LessEqual[y$46$re, 5e+69], N[(N[(x$46$im * N[(y$46$re / 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 < -1.7e12 or 5.00000000000000036e69 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 52.5

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

   4. Applied rewrites 52.5%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if -1.7e12 < y.re < 5.00000000000000036e69

   1. Initial program 72.3%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 78.7

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 80.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1700000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, \frac{x.re}{-y.re}, x.im\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 210000000000:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma x.im (/ y.re y.im) (- x.re)) y.im)))
   (if (<= y.im -1.65e+78)
     t_0
     (if (<= y.im 210000000000.0)
       (/ (- 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_im, (y_46_re / y_46_im), -x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.65e+78) {
		tmp = t_0;
	} else if (y_46_im <= 210000000000.0) {
		tmp = (x_46_im - ((x_46_re * y_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-x_46_re)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.65e+78)
		tmp = t_0;
	elseif (y_46_im <= 210000000000.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(x_46_re * y_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+78], t$95$0, If[LessEqual[y$46$im, 210000000000.0], 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 < -1.65e78 or 2.1e11 < y.im

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.2%

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

   if -1.65e78 < y.im < 2.1e11

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 210000000000:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -1.05e+81)
     t_0
     (if (<= y.im 6.1e+59) (/ (- 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 <= -1.05e+81) {
		tmp = t_0;
	} else if (y_46_im <= 6.1e+59) {
		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 <= (-1.05d+81)) then
        tmp = t_0
    else if (y_46im <= 6.1d+59) 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 <= -1.05e+81) {
		tmp = t_0;
	} else if (y_46_im <= 6.1e+59) {
		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 <= -1.05e+81:
		tmp = t_0
	elif y_46_im <= 6.1e+59:
		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(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.05e+81)
		tmp = t_0;
	elseif (y_46_im <= 6.1e+59)
		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 <= -1.05e+81)
		tmp = t_0;
	elseif (y_46_im <= 6.1e+59)
		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, -1.05e+81], t$95$0, If[LessEqual[y$46$im, 6.1e+59], 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 < -1.0499999999999999e81 or 6.09999999999999973e59 < y.im

   1. Initial program 42.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -1.0499999999999999e81 < y.im < 6.09999999999999973e59

   1. Initial program 77.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq 6.1 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 240000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -1.65e+78)
     t_0
     (if (<= y.im 240000000000.0) (/ 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 <= -1.65e+78) {
		tmp = t_0;
	} else if (y_46_im <= 240000000000.0) {
		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 <= (-1.65d+78)) then
        tmp = t_0
    else if (y_46im <= 240000000000.0d0) 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 <= -1.65e+78) {
		tmp = t_0;
	} else if (y_46_im <= 240000000000.0) {
		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 <= -1.65e+78:
		tmp = t_0
	elif y_46_im <= 240000000000.0:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.65e+78)
		tmp = t_0;
	elseif (y_46_im <= 240000000000.0)
		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 <= -1.65e+78)
		tmp = t_0;
	elseif (y_46_im <= 240000000000.0)
		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, -1.65e+78], t$95$0, If[LessEqual[y$46$im, 240000000000.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.65e78 or 2.4e11 < y.im

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -1.65e78 < y.im < 2.4e11

   1. Initial program 76.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 67.9

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

   5. Applied rewrites 67.9%

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

Alternative 8: 43.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.9%

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

3. Taylor expanded in y.re around inf

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

   1. lower-/.f64 46.0

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

5. Applied rewrites 46.0%

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

Reproduce

herbie shell --seed 2024237 
(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))))