math.cube on complex, imaginary part

Percentage Accurate: 82.0% → 99.7%

Time: 13.4s

Alternatives: 11

Speedup: 1.1×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.im\_m, -x.im\_m, x.re \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.re + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 2e+21)
      (* x.im_m (fma x.im_m (- x.im_m) (* x.re (* x.re 3.0))))
      (if (<= t_0 INFINITY)
        (* x.re (* (* x.re x.im_m) 3.0))
        (fma (- x.re x.im_m) (* x.im_m (+ x.re x.im_m)) (+ x.im_m x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e+21) {
		tmp = x_46_im_m * fma(x_46_im_m, -x_46_im_m, (x_46_re * (x_46_re * 3.0)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_re + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 2e+21)
		tmp = Float64(x_46_im_m * fma(x_46_im_m, Float64(-x_46_im_m), Float64(x_46_re * Float64(x_46_re * 3.0))));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_re + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e+21], N[(x$46$im$95$m * N[(x$46$im$95$m * (-x$46$im$95$m) + N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 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)) < 2e21

   1. Initial program 99.8%

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

   3. Taylor expanded in x.re around 0

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   if 2e21 < (+.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 78.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 10.6

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

   4. Applied rewrites 10.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 78.4

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

   9. Applied rewrites 78.4%

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

   11. Applied rewrites 99.8%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, x.re \cdot \left(x.re \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x.im\_m = \left|x.im\right| \\ x.im\_s = \mathsf{copysign}\left(1, x.im\right) \\ \begin{array}{l} t_0 := x.im\_m \cdot \left(x.re \cdot x.re - x.im\_m \cdot x.im\_m\right) + x.re \cdot \left(x.re \cdot x.im\_m + x.re \cdot x.im\_m\right)\\ x.im\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-282}:\\ \;\;\;\;x.im\_m \cdot \mathsf{fma}\left(x.im\_m, -x.im\_m, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\_m\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im\_m, x.im\_m \cdot \left(x.re + x.im\_m\right), x.im\_m + x.im\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x.im\_m = (fabs.f64 x.im)
x.im\_s = (copysign.f64 #s(literal 1 binary64) x.im)
(FPCore (x.im_s x.re x.im_m)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im_m (- (* x.re x.re) (* x.im_m x.im_m)))
          (* x.re (+ (* x.re x.im_m) (* x.re x.im_m))))))
   (*
    x.im_s
    (if (<= t_0 2e-282)
      (* x.im_m (fma x.im_m (- x.im_m) (* (* x.re x.re) 3.0)))
      (if (<= t_0 INFINITY)
        (* x.re (* (* x.re x.im_m) 3.0))
        (fma (- x.re x.im_m) (* x.im_m (+ x.re x.im_m)) (+ x.im_m x.im_m)))))))

C

x.im\_m = fabs(x_46_im);
x.im\_s = copysign(1.0, x_46_im);
double code(double x_46_im_s, double x_46_re, double x_46_im_m) {
	double t_0 = (x_46_im_m * ((x_46_re * x_46_re) - (x_46_im_m * x_46_im_m))) + (x_46_re * ((x_46_re * x_46_im_m) + (x_46_re * x_46_im_m)));
	double tmp;
	if (t_0 <= 2e-282) {
		tmp = x_46_im_m * fma(x_46_im_m, -x_46_im_m, ((x_46_re * x_46_re) * 3.0));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x_46_re * ((x_46_re * x_46_im_m) * 3.0);
	} else {
		tmp = fma((x_46_re - x_46_im_m), (x_46_im_m * (x_46_re + x_46_im_m)), (x_46_im_m + x_46_im_m));
	}
	return x_46_im_s * tmp;
}

Julia

x.im\_m = abs(x_46_im)
x.im\_s = copysign(1.0, x_46_im)
function code(x_46_im_s, x_46_re, x_46_im_m)
	t_0 = Float64(Float64(x_46_im_m * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im_m * x_46_im_m))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) + Float64(x_46_re * x_46_im_m))))
	tmp = 0.0
	if (t_0 <= 2e-282)
		tmp = Float64(x_46_im_m * fma(x_46_im_m, Float64(-x_46_im_m), Float64(Float64(x_46_re * x_46_re) * 3.0)));
	elseif (t_0 <= Inf)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im_m) * 3.0));
	else
		tmp = fma(Float64(x_46_re - x_46_im_m), Float64(x_46_im_m * Float64(x_46_re + x_46_im_m)), Float64(x_46_im_m + x_46_im_m));
	end
	return Float64(x_46_im_s * tmp)
end

Wolfram

x.im\_m = N[Abs[x$46$im], $MachinePrecision]
x.im\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x$46$im]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$46$im$95$s_, x$46$re_, x$46$im$95$m_] := Block[{t$95$0 = N[(N[(x$46$im$95$m * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im$95$m * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] + N[(x$46$re * x$46$im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$46$im$95$s * If[LessEqual[t$95$0, 2e-282], N[(x$46$im$95$m * N[(x$46$im$95$m * (-x$46$im$95$m) + N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x$46$re * N[(N[(x$46$re * x$46$im$95$m), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - x$46$im$95$m), $MachinePrecision] * N[(x$46$im$95$m * N[(x$46$re + x$46$im$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$46$im$95$m + x$46$im$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 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)) < 2e-282

   1. Initial program 99.9%

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

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

   if 2e-282 < (+.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 83.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 30.5

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

   4. Applied rewrites 30.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. difference-of-squares N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. associate-*r* N/A

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

   6. Applied rewrites 77.1%

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

   7. Taylor expanded in x.re around inf

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.0

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

   9. Applied rewrites 83.0%

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

   11. Applied rewrites 99.3%

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

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

   1. Initial program 0.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. difference-of-squares N/A

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

      6. lift--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. difference-of-squares N/A

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

      14. lower-*.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 0.0

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{-282}:\\ \;\;\;\;x.im \cdot \mathsf{fma}\left(x.im, -x.im, \left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x.re - x.im, x.im \cdot \left(x.re + x.im\right), x.im + x.im\right)\\ \end{array} \]
5. 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 2024226 
(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)))