math.cube on complex, real part

Percentage Accurate: 82.3% → 99.7%

Time: 6.4s

Precision: binary64

Cost: 2372

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

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

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.im) (* x.re (- x.re x.im)))))
   (if (<=
        (-
         (* x.re (- (* x.re x.re) (* x.im x.im)))
         (* x.im (+ (* x.re x.im) (* x.re x.im))))
        INFINITY)
     (- t_0 (* x.im (* x.im (+ x.re x.re))))
     t_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_im) * (x_46_re * (x_46_re - x_46_im));
	double tmp;
	if (((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_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = t_0 - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	} else {
		tmp = t_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_im) * (x_46_re * (x_46_re - x_46_im));
	double tmp;
	if (((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_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	} else {
		tmp = t_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_im) * (x_46_re * (x_46_re - x_46_im))
	tmp = 0
	if ((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_re * x_46_im)))) <= math.inf:
		tmp = t_0 - (x_46_im * (x_46_im * (x_46_re + x_46_re)))
	else:
		tmp = t_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(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(x_46_re - x_46_im)))
	tmp = 0.0
	if (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(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(x_46_im * Float64(x_46_re + x_46_re))));
	else
		tmp = t_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_im) * (x_46_re * (x_46_re - x_46_im));
	tmp = 0.0;
	if (((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_re * x_46_im)))) <= Inf)
		tmp = t_0 - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	else
		tmp = t_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[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$re * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 - N[(x$46$im * N[(x$46$im * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	82.3%
Target	86.8%
Herbie	99.7%

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

Derivation

1. Split input into 2 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 91.8%

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

   2. Simplified 91.8%

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

      [Start] 91.8%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 91.8%	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 91.8%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      *-commutative [<=] 91.8%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      distribute-lft-out [=>] 91.8%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 99.7%

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

      [Start] 91.8%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      sub-neg [=>] 91.8%	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
      *-commutative [=>] 91.8%	\[ \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      difference-of-squares [=>] 91.8%	\[ \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      associate-*l* [=>] 99.7%	\[ \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      *-commutative [=>] 99.7%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(-\color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im}\right) \]
      distribute-rgt-neg-in [=>] 99.7%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(-x.im\right)} \]
      distribute-rgt-in [=>] 99.7%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.im \cdot x.re + x.im \cdot x.re\right)} \cdot \left(-x.im\right) \]
      distribute-lft-out [=>] 99.7%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \cdot \left(-x.im\right) \]

   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. 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%

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

      [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%	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 0.0%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      *-commutative [<=] 0.0%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      distribute-lft-out [=>] 0.0%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 23.8%

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

      [Start] 0.0%	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      sub-neg [=>] 0.0%	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)} \]
      *-commutative [=>] 0.0%	\[ \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re} + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      difference-of-squares [=>] 23.8%	\[ \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.re + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      associate-*l* [=>] 23.8%	\[ \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} + \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right) \]
      *-commutative [=>] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(-\color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot x.im}\right) \]
      distribute-rgt-neg-in [=>] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \left(-x.im\right)} \]
      distribute-rgt-in [=>] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.im \cdot x.re + x.im \cdot x.re\right)} \cdot \left(-x.im\right) \]
      distribute-lft-out [=>] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \cdot \left(-x.im\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]
      Step-by-step derivation

      [Start] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.im \cdot \left(x.re + x.re\right)\right) \cdot \left(-x.im\right) \]
      distribute-lft-in [=>] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.im \cdot x.re + x.im \cdot x.re\right)} \cdot \left(-x.im\right) \]
      distribute-rgt-in [<=] 23.8%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot \left(-x.im\right) \]
      add-sqr-sqrt [=>] 0.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{\left(\sqrt{-x.im} \cdot \sqrt{-x.im}\right)} \]
      sqrt-unprod [=>] 52.4%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{\sqrt{\left(-x.im\right) \cdot \left(-x.im\right)}} \]
      sqr-neg [=>] 52.4%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \sqrt{\color{blue}{x.im \cdot x.im}} \]
      sqrt-prod [=>] 52.4%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)} \]
      add-sqr-sqrt [<=] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \left(x.re \cdot \left(x.im + x.im\right)\right) \cdot \color{blue}{x.im} \]
      *-commutative [<=] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      add-log-exp [=>] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{\log \left(e^{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)}\right)} \]
      associate-*r* [=>] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left(e^{\color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.im + x.im\right)}}\right) \]
      *-commutative [<=] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left(e^{\color{blue}{\left(x.re \cdot x.im\right)} \cdot \left(x.im + x.im\right)}\right) \]
      exp-prod [=>] 76.2%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \color{blue}{\left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + x.im\right)}\right)} \]
      add-sqr-sqrt [=>] 52.4%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + \color{blue}{\sqrt{x.im} \cdot \sqrt{x.im}}\right)}\right) \]
      sqrt-prod [<=] 61.9%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + \color{blue}{\sqrt{x.im \cdot x.im}}\right)}\right) \]
      sqr-neg [<=] 61.9%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + \sqrt{\color{blue}{\left(-x.im\right) \cdot \left(-x.im\right)}}\right)}\right) \]
      sqrt-unprod [<=] 14.3%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + \color{blue}{\sqrt{-x.im} \cdot \sqrt{-x.im}}\right)}\right) \]
      add-sqr-sqrt [<=] 100.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\left(x.im + \color{blue}{\left(-x.im\right)}\right)}\right) \]
      sub-neg [<=] 100.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\color{blue}{\left(x.im - x.im\right)}}\right) \]
      +-inverses [=>] 100.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \left({\left(e^{x.re \cdot x.im}\right)}^{\color{blue}{0}}\right) \]
      metadata-eval [=>] 100.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \log \color{blue}{1} \]
      metadata-eval [=>] 100.0%	\[ \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) + \color{blue}{0} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	2372

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

Alternative 2
Accuracy	96.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x.im \leq -7.6 \cdot 10^{+153} \lor \neg \left(x.im \leq 1.7 \cdot 10^{+153}\right):\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	82.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.7 \cdot 10^{+44} \lor \neg \left(x.im \leq 3.1 \cdot 10^{-15}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4
Accuracy	82.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.1 \cdot 10^{+44} \lor \neg \left(x.im \leq 1.25 \cdot 10^{-15}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5
Accuracy	58.4%
Cost	320

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

Reproduce

herbie shell --seed 2023165 
(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)))