math.cube on complex, real part

Average Accuracy: 88.2% → 99.2%

Time: 12.5s

Precision: binary64

Cost: 3656

Math

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

↓

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ t_1 := t_0 - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t_0 - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

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
 (let* ((t_0 (* x.re (- (* x.re x.re) (* x.im x.im))))
        (t_1 (- t_0 (* x.im (+ (* x.im x.re) (* x.im x.re))))))
   (if (<= t_1 (- INFINITY))
     (* -3.0 (* x.im (* x.im x.re)))
     (if (<= t_1 2e+296)
       (- t_0 (* 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 t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	} else if (t_1 <= 2e+296) {
		tmp = t_0 - (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;
}

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 t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	double t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	} else if (t_1 <= 2e+296) {
		tmp = t_0 - (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):
	t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re))
	elif t_1 <= 2e+296:
		tmp = t_0 - (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)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)))
	t_1 = Float64(t_0 - Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) + Float64(x_46_im * x_46_re))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_im * x_46_re)));
	elseif (t_1 <= 2e+296)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * 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)
	t_0 = x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	t_1 = t_0 - (x_46_im * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -3.0 * (x_46_im * (x_46_im * x_46_re));
	elseif (t_1 <= 2e+296)
		tmp = t_0 - (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_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-3.0 * N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+296], N[(t$95$0 - N[(x$46$im * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	88.2%
Target	99.6%
Herbie	99.2%

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

Derivation

1. Split input into 3 regimes

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

   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%

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

      [Start] 0.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      sub-neg [=>] 0.0	\[ \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      *-commutative [=>] 0.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      sub-neg [=>] 0.0	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      distribute-rgt-in [=>] 0.0	\[ \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.re + \left(-x.im \cdot x.im\right) \cdot x.re\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      associate-+l+ [=>] 0.0	\[ \color{blue}{\left(x.re \cdot x.re\right) \cdot x.re + \left(\left(-x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(\left(-x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re} \]
      *-commutative [=>] 0.0	\[ \left(\color{blue}{x.re \cdot \left(-x.im \cdot x.im\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-rgt-neg-in [=>] 0.0	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(-x.im\right)}\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 0.0	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \cdot \left(-x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-rgt-out [=>] 0.0	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot \left(-x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      associate-*l* [=>] 0.0	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot \left(-x.im\right)\right)}\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-lft-out [=>] 0.0	\[ \color{blue}{x.re \cdot \left(\left(-x.im \cdot x.im\right) + \left(x.im + x.im\right) \cdot \left(-x.im\right)\right)} + \left(x.re \cdot x.re\right) \cdot x.re \]
      fma-def [=>] 0.0	\[ \color{blue}{\mathsf{fma}\left(x.re, \left(-x.im \cdot x.im\right) + \left(x.im + x.im\right) \cdot \left(-x.im\right), \left(x.re \cdot x.re\right) \cdot x.re\right)} \]

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

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

   4. Simplified 99.4%

      \[\leadsto \color{blue}{-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)} \]
      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* [=>] 99.4	\[ -3 \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 99.4	\[ -3 \cdot \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -inf.0 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)) < 1.99999999999999996e296

   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	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      distribute-rgt-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 1.99999999999999996e296 < (-.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))

   1. Initial program 9.3%

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

   2. Simplified 9.4%

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

      [Start] 9.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      sub-neg [=>] 9.3	\[ \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      *-commutative [=>] 9.3	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      sub-neg [=>] 9.3	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      distribute-rgt-in [=>] 9.3	\[ \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot x.re + \left(-x.im \cdot x.im\right) \cdot x.re\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      associate-+l+ [=>] 9.3	\[ \color{blue}{\left(x.re \cdot x.re\right) \cdot x.re + \left(\left(-x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right)} \]
      +-commutative [=>] 9.3	\[ \color{blue}{\left(\left(-x.im \cdot x.im\right) \cdot x.re + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re} \]
      *-commutative [=>] 9.3	\[ \left(\color{blue}{x.re \cdot \left(-x.im \cdot x.im\right)} + \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-rgt-neg-in [=>] 9.3	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(-x.im\right)}\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 9.3	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \cdot \left(-x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-rgt-out [=>] 9.3	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot \left(-x.im\right)\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      associate-*l* [=>] 9.3	\[ \left(x.re \cdot \left(-x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot \left(-x.im\right)\right)}\right) + \left(x.re \cdot x.re\right) \cdot x.re \]
      distribute-lft-out [=>] 9.3	\[ \color{blue}{x.re \cdot \left(\left(-x.im \cdot x.im\right) + \left(x.im + x.im\right) \cdot \left(-x.im\right)\right)} + \left(x.re \cdot x.re\right) \cdot x.re \]
      fma-def [=>] 9.3	\[ \color{blue}{\mathsf{fma}\left(x.re, \left(-x.im \cdot x.im\right) + \left(x.im + x.im\right) \cdot \left(-x.im\right), \left(x.re \cdot x.re\right) \cdot x.re\right)} \]

   3. Taylor expanded in x.re around 0 2.5%

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

   4. Simplified 92.6%

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

      [Start] 2.5	\[ -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
      unpow2 [=>] 2.5	\[ -3 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.im\right)}\right) \]
      associate-*r* [=>] 92.6	\[ -3 \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 92.6	\[ -3 \cdot \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right)\right)} \]

   5. Taylor expanded in x.im around 0 2.5%

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

   6. Simplified 92.6%

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

      [Start] 2.5	\[ -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
      *-commutative [=>] 2.5	\[ -3 \cdot \color{blue}{\left({x.im}^{2} \cdot x.re\right)} \]
      unpow2 [=>] 2.5	\[ -3 \cdot \left(\color{blue}{\left(x.im \cdot x.im\right)} \cdot x.re\right) \]
      associate-*r* [<=] 92.6	\[ -3 \cdot \color{blue}{\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} \]
      *-commutative [=>] 92.6	\[ \color{blue}{\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3} \]
      associate-*r* [<=] 92.6	\[ \color{blue}{x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)} \]
      *-commutative [=>] 92.6	\[ x.im \cdot \color{blue}{\left(-3 \cdot \left(x.im \cdot x.re\right)\right)} \]
      *-commutative [=>] 92.6	\[ x.im \cdot \left(-3 \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq -\infty:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 2 \cdot 10^{+296}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7616

\[\mathsf{fma}\left(x.im, -2 \cdot \left(x.im \cdot x.re\right), \left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) + x.re \cdot 0\right) \]

Alternative 2
Accuracy	99.7%
Cost	7040

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

Alternative 3
Accuracy	99.6%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.im \leq -5.8 \cdot 10^{+92}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+106}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	1216

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

Alternative 5
Accuracy	99.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.8 \cdot 10^{+123}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+83}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 6
Accuracy	91.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x.im \leq -4.4 \cdot 10^{-75}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 8 \cdot 10^{-8}:\\ \;\;\;\;x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 7
Accuracy	91.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-75} \lor \neg \left(x.im \leq 8 \cdot 10^{-8}\right):\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 8
Accuracy	91.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 1.25 \cdot 10^{-7}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 9
Accuracy	58.0%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.55 \cdot 10^{-107} \lor \neg \left(x.re \leq 2.75 \cdot 10^{-104}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	55.8%
Cost	320

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

Reproduce

herbie shell --seed 2023137 
(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)))