math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 99.8%

Time: 9.5s

Alternatives: 10

Speedup: 1.3×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re - x.im\right)\\ \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, t\_0, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, t\_0, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (- x.re x.im))))
   (if (<=
        (+
         (* x.im (- (* x.re x.re) (* x.im x.im)))
         (* x.re (+ (* x.re x.im) (* x.re x.im))))
        INFINITY)
     (fma (+ x.re x.im) t_0 (* x.re (* x.re (+ x.im x.im))))
     (fma (+ x.re x.im) t_0 0.0))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_re - x_46_im);
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = fma((x_46_re + x_46_im), t_0, (x_46_re * (x_46_re * (x_46_im + x_46_im))));
	} else {
		tmp = fma((x_46_re + x_46_im), t_0, 0.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_re - x_46_im))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = fma(Float64(x_46_re + x_46_im), t_0, Float64(x_46_re * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = fma(Float64(x_46_re + x_46_im), t_0, 0.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re + x$46$im), $MachinePrecision] * t$95$0 + N[(x$46$re * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * t$95$0 + 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 92.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 99.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 37.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 37.9

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 37.9

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

   4. Applied rewrites 37.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0}}{x.im - x.im}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 - 0}}{x.im - x.im}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 \cdot 0} - 0}{x.im - x.im}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x.im - x.im}\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right) \]

      15. flip-- N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(0 - 0\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{0}\right) \]

      17. +-inverses N/A

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

      18. distribute-lft-out-- N/A

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

      19. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      20. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      21. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)}\right) \]

      22. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)}\right) \]

      23. +-inverses 100.0

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := \mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (fma (+ x.re x.im) (* x.im (- x.re x.im)) 0.0)))
   (if (<= t_0 -1e-313)
     t_1
     (if (<= t_0 INFINITY) (* 3.0 (* x.re (* x.re x.im))) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = fma((x_46_re + x_46_im), (x_46_im * (x_46_re - x_46_im)), 0.0);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_1 = fma(Float64(x_46_re + x_46_im), Float64(x_46_im * Float64(x_46_re - x_46_im)), 0.0)
	tmp = 0.0
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-313], t$95$1, If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000001e-313 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 72.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 86.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 86.8

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0}}{x.im - x.im}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 - 0}}{x.im - x.im}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 \cdot 0} - 0}{x.im - x.im}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x.im - x.im}\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right) \]

      15. flip-- N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(0 - 0\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{0}\right) \]

      17. +-inverses N/A

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

      18. distribute-lft-out-- N/A

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

      19. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      20. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      21. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)}\right) \]

      22. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)}\right) \]

      23. +-inverses 78.6

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]

   6. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]

   if -1.00000000001e-313 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      9. lower-*.f64 62.7

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \color{blue}{x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.0%

       \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-313}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= t_0 -1e-313)
     t_1
     (if (<= t_0 INFINITY) (* 3.0 (* x.re (* x.re x.im))) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if t_0 <= -1e-313:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-313], t$95$1, If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000001e-313 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 72.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 54.9

         \[\leadsto \left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

   if -1.00000000001e-313 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      9. lower-*.f64 62.7

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \color{blue}{x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 69.0%

       \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-313}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= t_0 -1e-313)
     t_1
     (if (<= t_0 INFINITY) (* (* x.re x.im) (* x.re 3.0)) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if t_0 <= -1e-313:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-313], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000001e-313 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 72.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 54.9

         \[\leadsto \left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

   if -1.00000000001e-313 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      9. lower-*.f64 62.7

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.re \cdot 3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-313}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= t_0 -1e-313)
     t_1
     (if (<= t_0 INFINITY) (* x.re (* x.im (* x.re 3.0))) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if t_0 <= -1e-313:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = x_46_re * (x_46_im * (x_46_re * 3.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-313], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000001e-313 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 72.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 54.9

         \[\leadsto \left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

   if -1.00000000001e-313 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 50.2%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      9. lower-*.f64 62.7

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 62.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \color{blue}{x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-313}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (* x.im (* x.im (- x.im)))))
   (if (<= t_0 -1e-313)
     t_1
     (if (<= t_0 INFINITY) (* x.im (* (* x.re x.re) 3.0)) t_1))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = x_46_im * (x_46_im * -x_46_im);
	double tmp;
	if (t_0 <= -1e-313) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_1 = x_46_im * (x_46_im * -x_46_im)
	tmp = 0
	if t_0 <= -1e-313:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	t_1 = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
	tmp = 0.0
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_1 = x_46_im * (x_46_im * -x_46_im);
	tmp = 0.0;
	if (t_0 <= -1e-313)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-313], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < -1.00000000001e-313 or +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 72.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

      2. unpow3 N/A

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

      3. unpow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 54.9

         \[\leadsto \left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]

   5. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

   if -1.00000000001e-313 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 93.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around inf

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

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -1 \cdot 10^{-313}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 5e+76)
   (fma (* x.im (* x.re 3.0)) x.re (* x.im (* x.im (- x.im))))
   (fma (+ x.re x.im) (* x.im (- x.re x.im)) 0.0)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 5e+76) {
		tmp = fma((x_46_im * (x_46_re * 3.0)), x_46_re, (x_46_im * (x_46_im * -x_46_im)));
	} else {
		tmp = fma((x_46_re + x_46_im), (x_46_im * (x_46_re - x_46_im)), 0.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 5e+76)
		tmp = fma(Float64(x_46_im * Float64(x_46_re * 3.0)), x_46_re, Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im))));
	else
		tmp = fma(Float64(x_46_re + x_46_im), Float64(x_46_im * Float64(x_46_re - x_46_im)), 0.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 5e+76], N[(N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] * x$46$re + N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.99999999999999991e76

   1. Initial program 86.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.im, -x.im, 3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.7%

      \[\leadsto \mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), \color{blue}{x.re}, x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\right) \]

   if 4.99999999999999991e76 < x.im

   1. Initial program 64.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 79.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.2

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-inverses N/A

         \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0}}{x.im - x.im}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 - 0}}{x.im - x.im}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{\color{blue}{0 \cdot 0} - 0}{x.im - x.im}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x.im - x.im}\right) \]

      13. +-inverses N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right) \]

      15. flip-- N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{\left(0 - 0\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.re\right) \cdot \color{blue}{0}\right) \]

      17. +-inverses N/A

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

      18. distribute-lft-out-- N/A

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

      19. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      20. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)} - \left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right)\right) \]

      21. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im\right) + 0\right)}\right) \]

      22. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right) - \color{blue}{\mathsf{fma}\left(x.re \cdot x.re, x.im \cdot x.im, 0\right)}\right) \]

      23. +-inverses 100.0

          \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]

   6. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x.re + x.im, \left(x.re - x.im\right) \cdot x.im, \color{blue}{0}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(x.im \cdot \left(x.re \cdot 3\right), x.re, x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.im \cdot \left(x.re - x.im\right), 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.4e+137)
   (* x.im (fma x.im (- x.im) (* (* x.re x.re) 3.0)))
   (fma (+ x.re x.im) (* x.re x.im) (* x.re (* x.re (+ x.im x.im))))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.4e+137) {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re * x_46_re) * 3.0));
	} else {
		tmp = fma((x_46_re + x_46_im), (x_46_re * x_46_im), (x_46_re * (x_46_re * (x_46_im + x_46_im))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.4e+137)
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	else
		tmp = fma(Float64(x_46_re + x_46_im), Float64(x_46_re * x_46_im), Float64(x_46_re * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.4e+137], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.4e137

   1. Initial program 87.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.im, -x.im, 3 \cdot \left(x.re \cdot x.re\right)\right)} \]

   if 1.4e137 < x.re

   1. Initial program 51.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. difference-of-squares N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 89.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.6

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 89.6

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

   7. Applied rewrites 89.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re + x.im, x.re \cdot x.im, x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.4e+137)
   (* x.im (fma x.im (- x.im) (* (* x.re x.re) 3.0)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.4e+137) {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re * x_46_re) * 3.0));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.4e+137)
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.4e+137], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.4e137

   1. Initial program 87.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{x.im \cdot \mathsf{fma}\left(x.im, -x.im, 3 \cdot \left(x.re \cdot x.re\right)\right)} \]

   if 1.4e137 < x.re

   1. Initial program 51.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

      9. lower-*.f64 58.9

         \[\leadsto x.im \cdot \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   7. Applied rewrites 58.9%

      \[\leadsto \color{blue}{x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.5%

      \[\leadsto \left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \color{blue}{x.re} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.6%

       \[\leadsto \left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+137}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(x.im \cdot \left(-x.im\right)\right) \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Fortran

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

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_im * -x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_im * -x_46_im);
end

Wolfram

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

Derivation

1. Initial program 82.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Add Preprocessing

3. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({x.im}^{3}\right)} \]

   2. unpow3 N/A

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

   3. unpow2 N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 57.0

      \[\leadsto \left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-x.im\right)} \]

5. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)} \]

6. Final simplification 57.0%

   \[\leadsto x.im \cdot \left(x.im \cdot \left(-x.im\right)\right) \]
7. Add Preprocessing

Developer Target 1: 91.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024219 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im))))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))