math.cube on complex, real part

Average Accuracy: 50.1% → 56.5%

Time: 10.7s

Precision: binary64

Cost: 9608

Math

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

↓

\[\begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \mathbf{elif}\;t_1 \leq 10^{+299}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \]

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.im (* x.im (* x.re -3.0))))
        (t_1
         (-
          (* x.re (- (* x.re x.re) (* x.im x.im)))
          (* x.im (+ (* x.re x.im) (* x.re x.im))))))
   (if (<= t_1 (- INFINITY))
     (+ (+ 1.0 t_0) -1.0)
     (if (<= t_1 1e+299)
       (+ (pow x.re 3.0) (* -3.0 (* x.re (* x.im x.im))))
       (fabs 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_im * (x_46_im * (x_46_re * -3.0));
	double t_1 = (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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (1.0 + t_0) + -1.0;
	} else if (t_1 <= 1e+299) {
		tmp = pow(x_46_re, 3.0) + (-3.0 * (x_46_re * (x_46_im * x_46_im)));
	} else {
		tmp = fabs(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_im * (x_46_im * (x_46_re * -3.0));
	double t_1 = (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 tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (1.0 + t_0) + -1.0;
	} else if (t_1 <= 1e+299) {
		tmp = Math.pow(x_46_re, 3.0) + (-3.0 * (x_46_re * (x_46_im * x_46_im)));
	} else {
		tmp = Math.abs(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_im * (x_46_im * (x_46_re * -3.0))
	t_1 = (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)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (1.0 + t_0) + -1.0
	elif t_1 <= 1e+299:
		tmp = math.pow(x_46_re, 3.0) + (-3.0 * (x_46_re * (x_46_im * x_46_im)))
	else:
		tmp = math.fabs(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(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 + t_0) + -1.0);
	elseif (t_1 <= 1e+299)
		tmp = Float64((x_46_re ^ 3.0) + Float64(-3.0 * Float64(x_46_re * Float64(x_46_im * x_46_im))));
	else
		tmp = abs(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_im * (x_46_im * (x_46_re * -3.0));
	t_1 = (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)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (1.0 + t_0) + -1.0;
	elseif (t_1 <= 1e+299)
		tmp = (x_46_re ^ 3.0) + (-3.0 * (x_46_re * (x_46_im * x_46_im)));
	else
		tmp = abs(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[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(1.0 + t$95$0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], N[(N[Power[x$46$re, 3.0], $MachinePrecision] + N[(-3.0 * N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[t$95$0], $MachinePrecision]]]]]

Target

Original	50.1%
Target	56.1%
Herbie	56.5%

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

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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \left(x.re \cdot x.im\right)\right)\right)} \]
      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 0.5%

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

   4. Simplified 16.9%

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

      [Start] 0.5	\[ \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right) \cdot x.re \]
      distribute-rgt-out [=>] 0.5	\[ \color{blue}{\left({x.im}^{2} \cdot \left(-2 + -1\right)\right)} \cdot x.re \]
      metadata-eval [=>] 0.5	\[ \left({x.im}^{2} \cdot \color{blue}{-3}\right) \cdot x.re \]
      metadata-eval [<=] 0.5	\[ \left({x.im}^{2} \cdot \color{blue}{\left(-1 + -2\right)}\right) \cdot x.re \]
      associate-*l* [=>] 0.5	\[ \color{blue}{{x.im}^{2} \cdot \left(\left(-1 + -2\right) \cdot x.re\right)} \]
      *-commutative [<=] 0.5	\[ {x.im}^{2} \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)} \]
      distribute-rgt-out [<=] 0.5	\[ {x.im}^{2} \cdot \color{blue}{\left(-1 \cdot x.re + -2 \cdot x.re\right)} \]
      unpow2 [=>] 0.5	\[ \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right) \]
      associate-*l* [=>] 16.9	\[ \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right)\right)} \]
      distribute-rgt-out [=>] 16.9	\[ x.im \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)}\right) \]
      metadata-eval [=>] 16.9	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot \color{blue}{-3}\right)\right) \]

   5. Applied egg-rr 17.0%

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

      [Start] 16.9	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
      expm1-log1p-u [=>] 0.4	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)\right)} \]
      expm1-udef [=>] 0.5	\[ \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)} - 1} \]
      log1p-udef [=>] 0.5	\[ e^{\color{blue}{\log \left(1 + x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)}} - 1 \]
      add-exp-log [<=] 17.0	\[ \color{blue}{\left(1 + x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)} - 1 \]

   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)) < 1.0000000000000001e299

   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 99.8%

      \[\leadsto \color{blue}{{x.re}^{3} + \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot -3} \]
      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	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      sub-neg [=>] 99.7	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      distribute-lft-in [=>] 99.7	\[ \color{blue}{\left(x.re \cdot \left(x.re \cdot x.re\right) + x.re \cdot \left(-x.im \cdot x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      associate--l+ [=>] 99.7	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re\right) + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      cube-unmult [=>] 99.8	\[ \color{blue}{{x.re}^{3}} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right) \]
      *-commutative [<=] 99.8	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im\right) \]
      distribute-lft-out [=>] 99.8	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im\right) \]
      associate-*l* [=>] 99.8	\[ {x.re}^{3} + \left(x.re \cdot \left(-x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)}\right) \]
      distribute-lft-out-- [=>] 99.7	\[ {x.re}^{3} + \color{blue}{x.re \cdot \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 99.7	\[ {x.re}^{3} + x.re \cdot \left(\color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 99.7	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 99.7	\[ {x.re}^{3} + x.re \cdot \left(-1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 99.7	\[ {x.re}^{3} + x.re \cdot \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)\right)} \]
      associate-*l* [<=] 99.8	\[ {x.re}^{3} + \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \left(-1 - 2\right)} \]
      metadata-eval [=>] 99.8	\[ {x.re}^{3} + \left(x.re \cdot \left(x.im \cdot x.im\right)\right) \cdot \color{blue}{-3} \]

   if 1.0000000000000001e299 < (-.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 0.6%

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

   2. Simplified 0.6%

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

      [Start] 0.6	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [=>] 0.6	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      fma-neg [=>] 0.6	\[ \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, -\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\right)} \]
      distribute-lft-neg-in [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{\left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right) \cdot x.im}\right) \]
      *-commutative [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, \color{blue}{x.im \cdot \left(-\left(x.re \cdot x.im + x.im \cdot x.re\right)\right)}\right) \]
      *-commutative [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)\right)\right) \]
      count-2 [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-\color{blue}{2 \cdot \left(x.im \cdot x.re\right)}\right)\right) \]
      distribute-lft-neg-in [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \color{blue}{\left(\left(-2\right) \cdot \left(x.im \cdot x.re\right)\right)}\right) \]
      metadata-eval [=>] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(\color{blue}{-2} \cdot \left(x.im \cdot x.re\right)\right)\right) \]
      *-commutative [<=] 0.6	\[ \mathsf{fma}\left(x.re, x.re \cdot x.re - x.im \cdot x.im, x.im \cdot \left(-2 \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right)\right) \]

   3. Taylor expanded in x.re around 0 0.4%

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

   4. Simplified 9.8%

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

      [Start] 0.4	\[ \left(-2 \cdot {x.im}^{2} + -1 \cdot {x.im}^{2}\right) \cdot x.re \]
      distribute-rgt-out [=>] 0.4	\[ \color{blue}{\left({x.im}^{2} \cdot \left(-2 + -1\right)\right)} \cdot x.re \]
      metadata-eval [=>] 0.4	\[ \left({x.im}^{2} \cdot \color{blue}{-3}\right) \cdot x.re \]
      metadata-eval [<=] 0.4	\[ \left({x.im}^{2} \cdot \color{blue}{\left(-1 + -2\right)}\right) \cdot x.re \]
      associate-*l* [=>] 0.4	\[ \color{blue}{{x.im}^{2} \cdot \left(\left(-1 + -2\right) \cdot x.re\right)} \]
      *-commutative [<=] 0.4	\[ {x.im}^{2} \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)} \]
      distribute-rgt-out [<=] 0.4	\[ {x.im}^{2} \cdot \color{blue}{\left(-1 \cdot x.re + -2 \cdot x.re\right)} \]
      unpow2 [=>] 0.4	\[ \color{blue}{\left(x.im \cdot x.im\right)} \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right) \]
      associate-*l* [=>] 9.8	\[ \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 \cdot x.re + -2 \cdot x.re\right)\right)} \]
      distribute-rgt-out [=>] 9.8	\[ x.im \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot \left(-1 + -2\right)\right)}\right) \]
      metadata-eval [=>] 9.8	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot \color{blue}{-3}\right)\right) \]

   5. Applied egg-rr 2.4%

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

      [Start] 9.8	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) \]
      add-sqr-sqrt [=>] 9.5	\[ \color{blue}{\sqrt{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)} \cdot \sqrt{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}} \]
      sqrt-unprod [=>] 2.4	\[ \color{blue}{\sqrt{\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)}} \]
      pow2 [=>] 2.4	\[ \sqrt{\color{blue}{{\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)}^{2}}} \]

   6. Applied egg-rr 10.6%

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

      [Start] 2.4	\[ \sqrt{{\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)}^{2}} \]
      unpow2 [=>] 2.4	\[ \sqrt{\color{blue}{\left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right) \cdot \left(x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right)}} \]
      rem-sqrt-square [=>] 10.6	\[ \color{blue}{\left|x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right|} \]

3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq -\infty:\\ \;\;\;\;\left(1 + x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right) + -1\\ \mathbf{elif}\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+299}:\\ \;\;\;\;{x.re}^{3} + -3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right|\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.2%
Cost	13312

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

Alternative 2
Accuracy	56.3%
Cost	9416

\[\begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{+302}:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \mathbf{elif}\;t_1 \leq 10^{+299}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right|\\ \end{array} \]

Alternative 3
Accuracy	56.0%
Cost	3273

\[\begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+302} \lor \neg \left(t_0 \leq 10^{+299}\right):\\ \;\;\;\;\left(1 + x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	55.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.9 \cdot 10^{+27}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{+67}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]

Alternative 5
Accuracy	45.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.4 \cdot 10^{-43} \lor \neg \left(x.im \leq 1.25 \cdot 10^{-38}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 6
Accuracy	51.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.9 \cdot 10^{-41} \lor \neg \left(x.im \leq 1.3 \cdot 10^{-38}\right):\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 7
Accuracy	51.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.55 \cdot 10^{-40} \lor \neg \left(x.im \leq 2.12 \cdot 10^{-38}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 8
Accuracy	51.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	51.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.im \leq -4.9 \cdot 10^{-45}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 1.75 \cdot 10^{-38}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \end{array} \]

Alternative 10
Accuracy	33.8%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.8 \cdot 10^{-102} \lor \neg \left(x.re \leq 5 \cdot 10^{-107}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 11
Accuracy	15.8%
Cost	320

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

Alternative 12
Accuracy	31.6%
Cost	320

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

Reproduce

herbie shell --seed 2023153 
(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)))