math.cube on complex, imaginary part

Percentage Accurate: 81.9% → 99.7%

Time: 10.8s

Alternatives: 15

Speedup: 1.3×

• 43.4% 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: 81.9% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(x.im, \left(x.re\_m + x.im\right) \cdot \left(x.re\_m - x.im\right), x.im + x.im\right)}}\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
       (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im))))
      INFINITY)
   (+
    (* (* x.im (+ x.re_m x.im)) (- x.re_m x.im))
    (* x.re_m (* x.re_m (+ x.im x.im))))
   (/
    1.0
    (/ 1.0 (fma x.im (* (+ x.re_m x.im) (- x.re_m x.im)) (+ x.im x.im))))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))) <= ((double) INFINITY)) {
		tmp = ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im)) + (x_46_re_m * (x_46_re_m * (x_46_im + x_46_im)));
	} else {
		tmp = 1.0 / (1.0 / fma(x_46_im, ((x_46_re_m + x_46_im) * (x_46_re_m - x_46_im)), (x_46_im + x_46_im)));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im)))) <= Inf)
		tmp = Float64(Float64(Float64(x_46_im * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)) + Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(x_46_im, Float64(Float64(x_46_re_m + x_46_im) * Float64(x_46_re_m - x_46_im)), Float64(x_46_im + x_46_im))));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$im * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x$46$im * N[(N[(x$46$re$95$m + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 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

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 93.5

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

   4. Applied egg-rr 99.8%

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

   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

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 40.9

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 40.9

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 50.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(x.im, \left(x.re + x.im\right) \cdot \left(x.re - x.im\right), x.im + x.im\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 2: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.im + x.im, x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY)
       (* x.re_m (fma x.re_m (+ x.im x.im) (* x.re_m x.im)))
       t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re_m * fma(x_46_re_m, (x_46_im + x_46_im), (x_46_re_m * x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re_m * fma(x_46_re_m, Float64(x_46_im + x_46_im), Float64(x_46_re_m * x_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 94.8

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 94.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 60.0

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

   7. Simplified 60.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 65.0

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

   9. Applied egg-rr 65.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.im + x.im, x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY) (* x.re_m (* x.re_m (* x.im 3.0))) t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = -(x_46_im * (x_46_im * x_46_im))
	tmp = 0
	if t_0 <= -5e-323:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0))
	else:
		tmp = t_1
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = -(x_46_im * (x_46_im * x_46_im));
	tmp = 0.0;
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = x_46_re_m * (x_46_re_m * (x_46_im * 3.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im * 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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im + 2 \cdot x.im\right)} \]
   4. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Simplified 60.0%

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

      1. lift-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 64.9

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

   7. Applied egg-rr 64.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x.re\_m \cdot x.im\right) \cdot \left(x.re\_m \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY) (* (* x.re_m x.im) (* x.re_m 3.0)) t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = -(x_46_im * (x_46_im * x_46_im))
	tmp = 0
	if t_0 <= -5e-323:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0)
	else:
		tmp = t_1
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_46_re_m * x_46_im) * Float64(x_46_re_m * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = -(x_46_im * (x_46_im * x_46_im));
	tmp = 0.0;
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = (x_46_re_m * x_46_im) * (x_46_re_m * 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] * N[(x$46$re$95$m * 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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im + 2 \cdot x.im\right)} \]
   4. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Simplified 60.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 64.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 64.9

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

   7. Applied egg-rr 64.9%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY) (* 3.0 (* (* x.re_m x.re_m) x.im)) t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = -(x_46_im * (x_46_im * x_46_im))
	tmp = 0
	if t_0 <= -5e-323:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im)
	else:
		tmp = t_1
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(3.0 * Float64(Float64(x_46_re_m * x_46_re_m) * x_46_im));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = -(x_46_im * (x_46_im * x_46_im));
	tmp = 0.0;
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = 3.0 * ((x_46_re_m * x_46_re_m) * x_46_im);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(3.0 * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im), $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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im + 2 \cdot x.im\right)} \]
   4. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Simplified 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 64.9

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 60.0

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

   7. Applied egg-rr 60.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.im \cdot \left(\left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY) (* x.im (* (* x.re_m x.re_m) 3.0)) t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_im * ((x_46_re_m * x_46_re_m) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_im * ((x_46_re_m * x_46_re_m) * 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = -(x_46_im * (x_46_im * x_46_im))
	tmp = 0
	if t_0 <= -5e-323:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = x_46_im * ((x_46_re_m * x_46_re_m) * 3.0)
	else:
		tmp = t_1
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) * 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = -(x_46_im * (x_46_im * x_46_im));
	tmp = 0.0;
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = x_46_im * ((x_46_re_m * x_46_re_m) * 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

   3. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{{x.re}^{2} \cdot \left(x.im + 2 \cdot x.im\right)} \]
   4. 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. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. *-rgt-identity N/A

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

      8. *-inverses N/A

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

      9. associate-/l* N/A

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

      10. unpow2 N/A

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

      11. cube-mult N/A

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

      12. associate-/l* N/A

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

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{2 \cdot {x.re}^{2} + {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3}} \]

      14. distribute-lft1-in N/A

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

      15. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{3} \cdot {x.re}^{2}}{{x.im}^{2}} \cdot {x.im}^{3} \]

      16. associate-*r/ N/A

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

      17. associate-*l* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

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

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

   5. Simplified 60.0%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re\_m \cdot x.re\_m - x.im \cdot x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot x.im + x.re\_m \cdot x.im\right)\\ t_1 := -x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re_m x.re_m) (* x.im x.im)))
          (* x.re_m (+ (* x.re_m x.im) (* x.re_m x.im)))))
        (t_1 (- (* x.im (* x.im x.im)))))
   (if (<= t_0 -5e-323)
     t_1
     (if (<= t_0 INFINITY) (* x.re_m (* x.re_m x.im)) t_1))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re_m * (x_46_re_m * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	double t_1 = -(x_46_im * (x_46_im * x_46_im));
	double tmp;
	if (t_0 <= -5e-323) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x_46_re_m * (x_46_re_m * x_46_im);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)))
	t_1 = -(x_46_im * (x_46_im * x_46_im))
	tmp = 0
	if t_0 <= -5e-323:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = x_46_re_m * (x_46_re_m * x_46_im)
	else:
		tmp = t_1
	return tmp

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re_m * x_46_re_m) - Float64(x_46_im * x_46_im))) + Float64(x_46_re_m * Float64(Float64(x_46_re_m * x_46_im) + Float64(x_46_re_m * x_46_im))))
	t_1 = Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re_m * Float64(x_46_re_m * x_46_im));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x.re_m = abs(x_46_re);
function tmp_2 = code(x_46_re_m, x_46_im)
	t_0 = (x_46_im * ((x_46_re_m * x_46_re_m) - (x_46_im * x_46_im))) + (x_46_re_m * ((x_46_re_m * x_46_im) + (x_46_re_m * x_46_im)));
	t_1 = -(x_46_im * (x_46_im * x_46_im));
	tmp = 0.0;
	if (t_0 <= -5e-323)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = x_46_re_m * (x_46_re_m * x_46_im);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(N[(x$46$re$95$m * x$46$im), $MachinePrecision] + N[(x$46$re$95$m * 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, -5e-323], t$95$1, If[LessEqual[t$95$0, Infinity], N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $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)) < -4.94066e-323 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 72.6%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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 57.6

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

   5. Simplified 57.6%

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

   if -4.94066e-323 < (+.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 94.8%

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

   3. Taylor expanded in x.re around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.0

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

   5. Simplified 60.0%

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

   7. Applied egg-rr 38.7%

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

   8. Taylor expanded in x.re around inf

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

      1. lower-*.f64 43.3

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

   10. Simplified 43.3%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -5 \cdot 10^{-323}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;-x.im \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(x.im, \frac{x.im}{x.re\_m}, x.im\right), x.re\_m - x.im, x.im \cdot \left(x.re\_m + x.re\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.re\_m \cdot 3, x.re\_m, x.im \cdot \left(-x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 5e-22)
   (*
    x.re_m
    (fma
     (fma x.im (/ x.im x.re_m) x.im)
     (- x.re_m x.im)
     (* x.im (+ x.re_m x.re_m))))
   (* x.im (fma (* x.re_m 3.0) x.re_m (* x.im (- x.im))))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 5e-22) {
		tmp = x_46_re_m * fma(fma(x_46_im, (x_46_im / x_46_re_m), x_46_im), (x_46_re_m - x_46_im), (x_46_im * (x_46_re_m + x_46_re_m)));
	} else {
		tmp = x_46_im * fma((x_46_re_m * 3.0), x_46_re_m, (x_46_im * -x_46_im));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 5e-22)
		tmp = Float64(x_46_re_m * fma(fma(x_46_im, Float64(x_46_im / x_46_re_m), x_46_im), Float64(x_46_re_m - x_46_im), Float64(x_46_im * Float64(x_46_re_m + x_46_re_m))));
	else
		tmp = Float64(x_46_im * fma(Float64(x_46_re_m * 3.0), x_46_re_m, Float64(x_46_im * Float64(-x_46_im))));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 5e-22], N[(x$46$re$95$m * N[(N[(x$46$im * N[(x$46$im / x$46$re$95$m), $MachinePrecision] + x$46$im), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision] + N[(x$46$im * N[(x$46$re$95$m + x$46$re$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re$95$m * 3.0), $MachinePrecision] * x$46$re$95$m + N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.99999999999999954e-22

   1. Initial program 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 89.2

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 89.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 97.2%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 97.2

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 95.3

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

   9. Simplified 95.3%

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

       1. lift-+.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift--.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. +-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. *-commutative N/A

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

       13. distribute-lft-out N/A

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

       14. lower-*.f64 N/A

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

       15. lower-fma.f64 93.4

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

       16. lift-*.f64 N/A

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

   11. Applied egg-rr 93.4%

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

   if 4.99999999999999954e-22 < x.im

   1. Initial program 79.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. Simplified 99.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-neg-out N/A

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

      13. lift-*.f64 N/A

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

      14. lower-neg.f64 98.2

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

   7. Applied egg-rr 98.2%

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

4. Final simplification 94.5%

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

Alternative 9: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\left(x.im \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right) + x.re\_m \cdot \left(x.re\_m \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 4e+56)
   (+
    (* (* x.im (+ x.re_m x.im)) (- x.re_m x.im))
    (* x.re_m (* x.re_m (+ x.im x.im))))
   (* x.im (fma x.im (- x.im) (* (* x.re_m x.re_m) 3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 4e+56) {
		tmp = ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im)) + (x_46_re_m * (x_46_re_m * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re_m * x_46_re_m) * 3.0));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4e+56)
		tmp = Float64(Float64(Float64(x_46_im * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)) + Float64(x_46_re_m * Float64(x_46_re_m * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 4e+56], N[(N[(N[(x$46$im * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.00000000000000037e56

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 87.9

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

   4. Applied egg-rr 95.5%

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

   if 4.00000000000000037e56 < x.im

   1. Initial program 74.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. Simplified 100.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im + x.im\right), x.re\_m, \left(x.im \cdot \left(x.re\_m + x.im\right)\right) \cdot \left(x.re\_m - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 4e+56)
   (fma
    (* x.re_m (+ x.im x.im))
    x.re_m
    (* (* x.im (+ x.re_m x.im)) (- x.re_m x.im)))
   (* x.im (fma x.im (- x.im) (* (* x.re_m x.re_m) 3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 4e+56) {
		tmp = fma((x_46_re_m * (x_46_im + x_46_im)), x_46_re_m, ((x_46_im * (x_46_re_m + x_46_im)) * (x_46_re_m - x_46_im)));
	} else {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re_m * x_46_re_m) * 3.0));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4e+56)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im + x_46_im)), x_46_re_m, Float64(Float64(x_46_im * Float64(x_46_re_m + x_46_im)) * Float64(x_46_re_m - x_46_im)));
	else
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 4e+56], N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + N[(N[(x$46$im * N[(x$46$re$95$m + x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re$95$m - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.00000000000000037e56

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 89.8

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 89.8

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 97.4%

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

   if 4.00000000000000037e56 < x.im

   1. Initial program 74.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. Simplified 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), x.re, \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.re\_m \cdot x.im\right), 3, -x.im \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.im 1.2e+57)
   (fma (* x.re_m (* x.re_m x.im)) 3.0 (- (* x.im (* x.im x.im))))
   (* x.im (fma x.im (- x.im) (* (* x.re_m x.re_m) 3.0)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.2e+57) {
		tmp = fma((x_46_re_m * (x_46_re_m * x_46_im)), 3.0, -(x_46_im * (x_46_im * x_46_im)));
	} else {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re_m * x_46_re_m) * 3.0));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.2e+57)
		tmp = fma(Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)), 3.0, Float64(-Float64(x_46_im * Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$im, 1.2e+57], N[(N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision] * 3.0 + (-N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.20000000000000002e57

   1. Initial program 87.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. Simplified 91.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-neg-out N/A

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

      18. lift-*.f64 N/A

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

      19. cube-mult N/A

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

      20. lower-neg.f64 N/A

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

      21. cube-mult N/A

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

      22. lift-*.f64 N/A

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

   7. Applied egg-rr 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. lower-fma.f64 87.4

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

   9. Applied egg-rr 94.5%

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

   if 1.20000000000000002e57 < x.im

   1. Initial program 74.5%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   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. Simplified 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.2 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.re \cdot x.im\right), 3, -x.im \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re\_m \cdot \left(x.im + x.im\right), x.re\_m, x.re\_m \cdot \left(x.re\_m \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 2.2e+149)
   (* x.im (fma x.im (- x.im) (* (* x.re_m x.re_m) 3.0)))
   (fma (* x.re_m (+ x.im x.im)) x.re_m (* x.re_m (* x.re_m x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.2e+149) {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re_m * x_46_re_m) * 3.0));
	} else {
		tmp = fma((x_46_re_m * (x_46_im + x_46_im)), x_46_re_m, (x_46_re_m * (x_46_re_m * x_46_im)));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.2e+149)
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)));
	else
		tmp = fma(Float64(x_46_re_m * Float64(x_46_im + x_46_im)), x_46_re_m, Float64(x_46_re_m * Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 2.2e+149], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$re$95$m + N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.2e149

   1. Initial program 90.0%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 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 2.2e149 < x.re

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 45.6

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 45.6

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 53.3

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

   7. Simplified 53.3%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-*.f64 92.1

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re \cdot \left(x.im + x.im\right), x.re, x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \begin{array}{l} \mathbf{if}\;x.re\_m \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re\_m \cdot x.re\_m\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re\_m \cdot \mathsf{fma}\left(x.re\_m, x.im + x.im, x.re\_m \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im)
 :precision binary64
 (if (<= x.re_m 2.4e+149)
   (* x.im (fma x.im (- x.im) (* (* x.re_m x.re_m) 3.0)))
   (* x.re_m (fma x.re_m (+ x.im x.im) (* x.re_m x.im)))))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	double tmp;
	if (x_46_re_m <= 2.4e+149) {
		tmp = x_46_im * fma(x_46_im, -x_46_im, ((x_46_re_m * x_46_re_m) * 3.0));
	} else {
		tmp = x_46_re_m * fma(x_46_re_m, (x_46_im + x_46_im), (x_46_re_m * x_46_im));
	}
	return tmp;
}

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	tmp = 0.0
	if (x_46_re_m <= 2.4e+149)
		tmp = Float64(x_46_im * fma(x_46_im, Float64(-x_46_im), Float64(Float64(x_46_re_m * x_46_re_m) * 3.0)));
	else
		tmp = Float64(x_46_re_m * fma(x_46_re_m, Float64(x_46_im + x_46_im), Float64(x_46_re_m * x_46_im)));
	end
	return tmp
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := If[LessEqual[x$46$re$95$m, 2.4e+149], N[(x$46$im * N[(x$46$im * (-x$46$im) + N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re$95$m * N[(x$46$re$95$m * N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.40000000000000012e149

   1. Initial program 90.0%

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

   3. Taylor expanded in x.re around 0

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

   5. Simplified 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 2.40000000000000012e149 < x.re

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 45.6

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-+.f64 45.6

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

   4. Applied egg-rr 92.1%

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

   5. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 53.3

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

   7. Simplified 53.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 92.0

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

   9. Applied egg-rr 92.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.4 \cdot 10^{+149}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \mathsf{fma}\left(x.re, x.im + x.im, x.re \cdot x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ x.re\_m \cdot \left(x.re\_m \cdot x.im\right) \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* x.re_m (* x.re_m x.im)))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return x_46_re_m * (x_46_re_m * x_46_im);
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = x_46re_m * (x_46re_m * x_46im)
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return x_46_re_m * (x_46_re_m * x_46_im);
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return x_46_re_m * (x_46_re_m * x_46_im)

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(x_46_re_m * Float64(x_46_re_m * x_46_im))
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = x_46_re_m * (x_46_re_m * x_46_im);
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(x$46$re$95$m * N[(x$46$re$95$m * x$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in x.re around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 50.8

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

5. Simplified 50.8%

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

7. Applied egg-rr 29.4%

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

8. Taylor expanded in x.re around inf

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

   1. lower-*.f64 32.2

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

10. Simplified 32.2%

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

11. Final simplification 32.2%

    \[\leadsto x.re \cdot \left(x.re \cdot x.im\right) \]
12. Add Preprocessing

Alternative 15: 34.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} x.re_m = \left|x.re\right| \\ \left(x.re\_m \cdot x.re\_m\right) \cdot x.im \end{array} \]

FPCore

x.re_m = (fabs.f64 x.re)
(FPCore (x.re_m x.im) :precision binary64 (* (* x.re_m x.re_m) x.im))

C

x.re_m = fabs(x_46_re);
double code(double x_46_re_m, double x_46_im) {
	return (x_46_re_m * x_46_re_m) * x_46_im;
}

Fortran

x.re_m = abs(x_46re)
real(8) function code(x_46re_m, x_46im)
    real(8), intent (in) :: x_46re_m
    real(8), intent (in) :: x_46im
    code = (x_46re_m * x_46re_m) * x_46im
end function

Java

x.re_m = Math.abs(x_46_re);
public static double code(double x_46_re_m, double x_46_im) {
	return (x_46_re_m * x_46_re_m) * x_46_im;
}

Python

x.re_m = math.fabs(x_46_re)
def code(x_46_re_m, x_46_im):
	return (x_46_re_m * x_46_re_m) * x_46_im

Julia

x.re_m = abs(x_46_re)
function code(x_46_re_m, x_46_im)
	return Float64(Float64(x_46_re_m * x_46_re_m) * x_46_im)
end

MATLAB

x.re_m = abs(x_46_re);
function tmp = code(x_46_re_m, x_46_im)
	tmp = (x_46_re_m * x_46_re_m) * x_46_im;
end

Wolfram

x.re_m = N[Abs[x$46$re], $MachinePrecision]
code[x$46$re$95$m_, x$46$im_] := N[(N[(x$46$re$95$m * x$46$re$95$m), $MachinePrecision] * x$46$im), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in x.re around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 50.8

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

5. Simplified 50.8%

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

7. Applied egg-rr 29.4%

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

8. Taylor expanded in x.re around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 31.6

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

10. Simplified 31.6%

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

11. Final simplification 31.6%

    \[\leadsto \left(x.re \cdot x.re\right) \cdot x.im \]
12. Add Preprocessing

Developer Target 1: 91.3% 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 2024207 
(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)))