math.cube on complex, real part

Average Accuracy: 88.4% → 99.6%

Time: 8.1s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -5e+154)
   (* (* x.im -3.0) (* x.re x.im))
   (if (<= x.im 4e+86)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* x.re (+ x.im x.im))))
     (* x.im (* x.im (* x.re -3.0))))))

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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5e+154) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 4e+86) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

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_46re) - (((x_46re * x_46im) + (x_46im * x_46re)) * x_46im)
end function

↓

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-5d+154)) then
        tmp = (x_46im * (-3.0d0)) * (x_46re * x_46im)
    else if (x_46im <= 4d+86) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    end if
    code = tmp
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -5e+154) {
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	} else if (x_46_im <= 4e+86) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	}
	return tmp;
}

Python

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

↓

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -5e+154:
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im)
	elif x_46_im <= 4e+86:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	return tmp

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_re) - Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_im))
end

↓

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -5e+154)
		tmp = Float64(Float64(x_46_im * -3.0) * Float64(x_46_re * x_46_im));
	elseif (x_46_im <= 4e+86)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	end
	return tmp
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_re) - (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_im);
end

↓

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -5e+154)
		tmp = (x_46_im * -3.0) * (x_46_re * x_46_im);
	elseif (x_46_im <= 4e+86)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - (x_46_im * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	end
	tmp_2 = tmp;
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$46$re), $MachinePrecision] - N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -5e+154], N[(N[(x$46$im * -3.0), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4e+86], N[(N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.4%
Target	99.6%
Herbie	99.6%

\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right) \]

Derivation

1. Split input into 3 regimes

2. if x.im < -5.00000000000000004e154

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 0.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      fma-neg [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, -\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      distribute-lft-neg-in [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{\left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right) \cdot x.im}\right) \]
      *-commutative [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{x.im \cdot \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)}\right) \]
      *-commutative [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)\right)\right) \]
      count-2 [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\color{blue}{2 \cdot \left(x.im \cdot x.re\right)}\right)\right) \]
      distribute-lft-neg-in [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \color{blue}{\left(\left(-2\right) \cdot \left(x.im \cdot x.re\right)\right)}\right) \]
      metadata-eval [=>] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(\color{blue}{-2} \cdot \left(x.im \cdot x.re\right)\right)\right) \]
      *-commutative [<=] 0.0	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

   3. Taylor expanded in x.re around 0 0.0%

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

   4. Simplified 0.0%

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

      [Start] 0.0	\[ \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right) \cdot x.re \]
      *-commutative [=>] 0.0	\[ \color{blue}{x.re \cdot \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right)} \]
      distribute-rgt-out [=>] 0.0	\[ x.re \cdot \color{blue}{\left({x.im}^{2} \cdot \left(-2 + -1\right)\right)} \]
      unpow2 [=>] 0.0	\[ x.re \cdot \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-2 + -1\right)\right) \]
      metadata-eval [=>] 0.0	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot \color{blue}{-3}\right) \]
      metadata-eval [<=] 0.0	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot \color{blue}{\left(-1 + -2\right)}\right) \]
      associate-*l* [=>] 0.0	\[ x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot \left(-1 + -2\right)\right)\right)} \]
      metadata-eval [=>] 0.0	\[ x.re \cdot \left(x.im \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \]

   5. Taylor expanded in x.re around 0 0.0%

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

   6. Simplified 99.4%

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

      [Start] 0.0	\[ -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
      unpow2 [=>] 0.0	\[ -3 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]
      associate-*r* [=>] 99.5	\[ -3 \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot x.im\right)} \]
      *-commutative [<=] 99.5	\[ -3 \cdot \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} \]
      associate-*r* [=>] 99.4	\[ \color{blue}{\left(-3 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

   if -5.00000000000000004e154 < x.im < 4.0000000000000001e86

   1. Initial program 99.7%

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

   2. Simplified 99.7%

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

      [Start] 99.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 99.7	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 99.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      *-commutative [<=] 99.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      distribute-lft-out [=>] 99.7	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   if 4.0000000000000001e86 < x.im

   1. Initial program 50.4%

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

   2. Simplified 50.4%

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

      [Start] 50.4	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 50.4	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      fma-neg [=>] 50.4	\[ \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, -\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      distribute-lft-neg-in [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{\left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right) \cdot x.im}\right) \]
      *-commutative [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{x.im \cdot \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)}\right) \]
      *-commutative [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)\right)\right) \]
      count-2 [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\color{blue}{2 \cdot \left(x.im \cdot x.re\right)}\right)\right) \]
      distribute-lft-neg-in [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \color{blue}{\left(\left(-2\right) \cdot \left(x.im \cdot x.re\right)\right)}\right) \]
      metadata-eval [=>] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(\color{blue}{-2} \cdot \left(x.im \cdot x.re\right)\right)\right) \]
      *-commutative [<=] 50.4	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

   3. Taylor expanded in x.re around 0 49.7%

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

   4. Simplified 99.2%

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

      [Start] 49.7	\[ \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right) \cdot x.re \]
      distribute-rgt-out [=>] 49.7	\[ \color{blue}{\left({x.im}^{2} \cdot \left(-2 + -1\right)\right)} \cdot x.re \]
      metadata-eval [=>] 49.7	\[ \left({x.im}^{2} \cdot \color{blue}{-3}\right) \cdot x.re \]
      metadata-eval [<=] 49.7	\[ \left({x.im}^{2} \cdot \color{blue}{\left(-1 + -2\right)}\right) \cdot x.re \]
      associate-*l* [=>] 50.1	\[ \color{blue}{{x.im}^{2} \cdot \left(\left(-1 + -2\right) \cdot x.re\right)} \]
      *-commutative [<=] 50.1	\[ {x.im}^{2} \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)} \]
      distribute-rgt-out [<=] 50.1	\[ {x.im}^{2} \cdot \color{blue}{\left(-1 \cdot x.re + -2 \cdot x.re\right)} \]
      unpow2 [=>] 50.1	\[ \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right) \]
      associate-*l* [=>] 99.2	\[ \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right)\right)} \]
      distribute-rgt-out [=>] 99.2	\[ x.im \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)}\right) \]
      metadata-eval [=>] 99.2	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot \color{blue}{-3}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+98}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right) + -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	1088

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

Alternative 3
Accuracy	79.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \cdot 10^{-33} \lor \neg \left(x.im \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4
Accuracy	91.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.1 \cdot 10^{-33} \lor \neg \left(x.im \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5
Accuracy	91.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.75 \cdot 10^{-27}:\\ \;\;\;\;\left(x.im \cdot -3\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{-18}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	55.2%
Cost	320

\[x.re \cdot \left(x.re \cdot x.re\right) \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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