_divideComplex, real part

Percentage Accurate: 62.0% → 82.4%

Time: 7.5s

Alternatives: 8

Speedup: 1.6×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{\mathsf{fma}\left(\frac{-y.re}{y.im}, x.im, x.re\right)}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.9 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 0.105:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re)))
        (t_1 (fma x.im (/ y.im t_0) (* y.re (/ x.re t_0)))))
   (if (<= y.im -1.6e+140)
     (fma
      (/ y.re y.im)
      (/ (fma (/ (- y.re) y.im) x.im x.re) y.im)
      (/ x.im y.im))
     (if (<= y.im -4.9e+26)
       t_1
       (if (<= y.im 0.105)
         (/ (fma (/ y.im y.re) x.im x.re) y.re)
         (if (<= y.im 4.6e+150)
           t_1
           (/ (fma (/ y.re y.im) x.re x.im) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double t_1 = fma(x_46_im, (y_46_im / t_0), (y_46_re * (x_46_re / t_0)));
	double tmp;
	if (y_46_im <= -1.6e+140) {
		tmp = fma((y_46_re / y_46_im), (fma((-y_46_re / y_46_im), x_46_im, x_46_re) / y_46_im), (x_46_im / y_46_im));
	} else if (y_46_im <= -4.9e+26) {
		tmp = t_1;
	} else if (y_46_im <= 0.105) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 4.6e+150) {
		tmp = t_1;
	} else {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	t_1 = fma(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / t_0)))
	tmp = 0.0
	if (y_46_im <= -1.6e+140)
		tmp = fma(Float64(y_46_re / y_46_im), Float64(fma(Float64(Float64(-y_46_re) / y_46_im), x_46_im, x_46_re) / y_46_im), Float64(x_46_im / y_46_im));
	elseif (y_46_im <= -4.9e+26)
		tmp = t_1;
	elseif (y_46_im <= 0.105)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 4.6e+150)
		tmp = t_1;
	else
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+140], N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(N[(N[((-y$46$re) / y$46$im), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.9e+26], t$95$1, If[LessEqual[y$46$im, 0.105], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.6e+150], t$95$1, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.60000000000000005e140

   1. Initial program 26.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 30.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 30.4

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(-1 \cdot \frac{x.im \cdot y.re}{{y.im}^{3}} + \frac{x.re}{{y.im}^{2}}\right) + \frac{x.im}{y.im}} \]

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 89.5%

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

   if -1.60000000000000005e140 < y.im < -4.89999999999999974e26 or 0.104999999999999996 < y.im < 4.60000000000000002e150

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 92.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if -4.89999999999999974e26 < y.im < 0.104999999999999996

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.5

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

   7. Applied rewrites 88.5%

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

   if 4.60000000000000002e150 < y.im

   1. Initial program 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 32.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 32.0

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

   4. Applied rewrites 32.0%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

Alternative 2: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ t_1 := \mathsf{fma}\left(x.im, \frac{y.im}{t\_0}, y.re \cdot \frac{x.re}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -4.9 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 0.105:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+150}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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.im (/ y.im t_0) (* y.re (/ x.re t_0))))
        (t_2 (/ (fma (/ y.re y.im) x.re x.im) y.im)))
   (if (<= y.im -1.6e+140)
     t_2
     (if (<= y.im -4.9e+26)
       t_1
       (if (<= y.im 0.105)
         (/ (fma (/ y.im y.re) x.im x.re) y.re)
         (if (<= y.im 4.6e+150) 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_im, (y_46_im / t_0), (y_46_re * (x_46_re / t_0)));
	double t_2 = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	double tmp;
	if (y_46_im <= -1.6e+140) {
		tmp = t_2;
	} else if (y_46_im <= -4.9e+26) {
		tmp = t_1;
	} else if (y_46_im <= 0.105) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 4.6e+150) {
		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(x_46_im, Float64(y_46_im / t_0), Float64(y_46_re * Float64(x_46_re / t_0)))
	t_2 = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.6e+140)
		tmp = t_2;
	elseif (y_46_im <= -4.9e+26)
		tmp = t_1;
	elseif (y_46_im <= 0.105)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 4.6e+150)
		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$46$im * N[(y$46$im / t$95$0), $MachinePrecision] + N[(y$46$re * N[(x$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+140], t$95$2, If[LessEqual[y$46$im, -4.9e+26], t$95$1, If[LessEqual[y$46$im, 0.105], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.6e+150], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.60000000000000005e140 or 4.60000000000000002e150 < y.im

   1. Initial program 26.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 31.2

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.0

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

   7. Applied rewrites 90.0%

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

   if -1.60000000000000005e140 < y.im < -4.89999999999999974e26 or 0.104999999999999996 < y.im < 4.60000000000000002e150

   1. Initial program 73.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 92.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if -4.89999999999999974e26 < y.im < 0.104999999999999996

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.5

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

   7. Applied rewrites 88.5%

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

Alternative 3: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4e+59)
   (/ x.im y.im)
   (if (<= y.im 6.2e+14)
     (/ (fma (/ y.im y.re) x.im x.re) y.re)
     (if (<= y.im 2.85e+144)
       (* (/ y.im (fma y.re y.re (* y.im y.im))) x.im)
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4e+59) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 6.2e+14) {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	} else if (y_46_im <= 2.85e+144) {
		tmp = (y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4e+59)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 6.2e+14)
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	elseif (y_46_im <= 2.85e+144)
		tmp = Float64(Float64(y_46_im / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4e+59], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 6.2e+14], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.85e+144], N[(N[(y$46$im / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.99999999999999989e59 or 2.85000000000000002e144 < y.im

   1. Initial program 34.3%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -3.99999999999999989e59 < y.im < 6.2e14

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 68.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 85.3

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

   7. Applied rewrites 85.3%

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

   if 6.2e14 < y.im < 2.85000000000000002e144

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 90.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.9

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

   7. Applied rewrites 57.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1050000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.02e+27)
   (/ x.im y.im)
   (if (<= y.im 1050000.0)
     (/ x.re y.re)
     (if (<= y.im 2.85e+144)
       (* (/ y.im (fma y.re y.re (* y.im y.im))) x.im)
       (/ x.im y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.02e+27) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1050000.0) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.85e+144) {
		tmp = (y_46_im / fma(y_46_re, y_46_re, (y_46_im * y_46_im))) * x_46_im;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.02e+27)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1050000.0)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 2.85e+144)
		tmp = Float64(Float64(y_46_im / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im))) * x_46_im);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.02e+27], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1050000.0], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.85e+144], N[(N[(y$46$im / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.0199999999999999e27 or 2.85000000000000002e144 < y.im

   1. Initial program 38.8%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   if -1.0199999999999999e27 < y.im < 1.05e6

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if 1.05e6 < y.im < 2.85000000000000002e144

   1. Initial program 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 90.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in x.re around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.9

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

   7. Applied rewrites 57.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1050000:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.85 \cdot 10^{+144}:\\ \;\;\;\;\frac{y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.7\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.4e+26) (not (<= y.im 1.7)))
   (/ (fma (/ x.re y.im) y.re x.im) y.im)
   (/ (fma (/ y.im y.re) x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.4e+26) || !(y_46_im <= 1.7)) {
		tmp = fma((x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9.4e+26) || !(y_46_im <= 1.7))
		tmp = Float64(fma(Float64(x_46_re / y_46_im), y_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.4e+26], N[Not[LessEqual[y$46$im, 1.7]], $MachinePrecision]], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] * y$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.3999999999999995e26 or 1.69999999999999996 < y.im

   1. Initial program 48.5%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -9.3999999999999995e26 < y.im < 1.69999999999999996

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.5

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

   7. Applied rewrites 88.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.7\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.7\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.4e+26) (not (<= y.im 1.7)))
   (/ (fma (/ y.re y.im) x.re x.im) y.im)
   (/ (fma (/ y.im y.re) x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.4e+26) || !(y_46_im <= 1.7)) {
		tmp = fma((y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im;
	} else {
		tmp = fma((y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9.4e+26) || !(y_46_im <= 1.7))
		tmp = Float64(fma(Float64(y_46_re / y_46_im), x_46_re, x_46_im) / y_46_im);
	else
		tmp = Float64(fma(Float64(y_46_im / y_46_re), x_46_im, x_46_re) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.4e+26], N[Not[LessEqual[y$46$im, 1.7]], $MachinePrecision]], N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re + x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(N[(y$46$im / y$46$re), $MachinePrecision] * x$46$im + x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.3999999999999995e26 or 1.69999999999999996 < y.im

   1. Initial program 48.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 59.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 59.4

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 79.6

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

   7. Applied rewrites 79.6%

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

   if -9.3999999999999995e26 < y.im < 1.69999999999999996

   1. Initial program 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 67.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.5

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

   7. Applied rewrites 88.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.4 \cdot 10^{+26} \lor \neg \left(y.im \leq 1.7\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.re}{y.im}, x.re, x.im\right)}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+27} \lor \neg \left(y.im \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.02e+27) (not (<= y.im 6.2e+14)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.02e+27) || !(y_46_im <= 6.2e+14)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.02d+27)) .or. (.not. (y_46im <= 6.2d+14))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.02e+27) || !(y_46_im <= 6.2e+14)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.02e+27) or not (y_46_im <= 6.2e+14):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.02e+27) || !(y_46_im <= 6.2e+14))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.02e+27) || ~((y_46_im <= 6.2e+14)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.02e+27], N[Not[LessEqual[y$46$im, 6.2e+14]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.0199999999999999e27 or 6.2e14 < y.im

   1. Initial program 48.0%

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

   3. Taylor expanded in y.re around 0

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

      1. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if -1.0199999999999999e27 < y.im < 6.2e14

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.02 \cdot 10^{+27} \lor \neg \left(y.im \leq 6.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_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_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 59.1%

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

3. Taylor expanded in y.re around 0

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

   1. lower-/.f64 39.4

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

5. Applied rewrites 39.4%

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

6. Final simplification 39.4%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))