math.cube on complex, real part

Average Error: 7.5 → 0.3

Time: 14.7s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+108}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \end{array} \]

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+108)
   (* x.im (* x.re (* x.im -3.0)))
   (if (<= x.im 6.6e+92)
     (* x.re (- (* (- x.re x.im) (+ x.re x.im)) (* x.im (+ x.im x.im))))
     (/ (* x.im (* x.im (* x.re -3.0))) 1.0))))

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+108) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 6.6e+92) {
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_re + x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	}
	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+108)) then
        tmp = x_46im * (x_46re * (x_46im * (-3.0d0)))
    else if (x_46im <= 6.6d+92) then
        tmp = x_46re * (((x_46re - x_46im) * (x_46re + x_46im)) - (x_46im * (x_46im + x_46im)))
    else
        tmp = (x_46im * (x_46im * (x_46re * (-3.0d0)))) / 1.0d0
    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+108) {
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	} else if (x_46_im <= 6.6e+92) {
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_re + x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	} else {
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	}
	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+108:
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0))
	elif x_46_im <= 6.6e+92:
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_re + x_46_im)) - (x_46_im * (x_46_im + x_46_im)))
	else:
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0
	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+108)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(x_46_im * -3.0)));
	elseif (x_46_im <= 6.6e+92)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_re + x_46_im)) - Float64(x_46_im * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0))) / 1.0);
	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+108)
		tmp = x_46_im * (x_46_re * (x_46_im * -3.0));
	elseif (x_46_im <= 6.6e+92)
		tmp = x_46_re * (((x_46_re - x_46_im) * (x_46_re + x_46_im)) - (x_46_im * (x_46_im + x_46_im)));
	else
		tmp = (x_46_im * (x_46_im * (x_46_re * -3.0))) / 1.0;
	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+108], N[(x$46$im * N[(x$46$re * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6.6e+92], N[(x$46$re * N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Target

Original	7.5
Target	0.2
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x.im < -4.99999999999999991e108

   1. Initial program 38.8

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

   2. Simplified 38.8

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

      [Start] 38.8	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 38.8	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 38.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-2 [<=] 38.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-47 [=>] 38.8	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-44 [=>] 38.9	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-2 [=>] 38.9	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re} \]
      rational_best-simplify-48 [=>] 38.8	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 38.8

      \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0} \]

   4. Simplified 38.9

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

      [Start] 38.8	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0 \]
      rational_best-simplify-4 [=>] 38.8	\[ \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right)} \]
      rational_best-simplify-46 [=>] 38.8	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(x.im \cdot x.im\right) \cdot -2 - x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-2 [=>] 38.8	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{-2 \cdot \left(x.im \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best-simplify-44 [=>] 38.8	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{x.im \cdot \left(-2 \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best-simplify-48 [=>] 38.9	\[ x.re \cdot \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(-2 \cdot x.im - x.im\right)}\right) \]
      rational_best-simplify-2 [=>] 38.9	\[ x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(\color{blue}{x.im \cdot -2} - x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(-2 \cdot x.im - x.im\right) \cdot \left(x.re \cdot x.im\right) \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(-2 \cdot x.im - x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)} \]
      rational_best-simplify-44 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(\left(-2 \cdot x.im - x.im\right) \cdot x.re\right)} \]

   7. Applied egg-rr 0.4

      \[\leadsto \color{blue}{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) + 0} \]

   8. Simplified 0.4

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

      [Start] 0.4	\[ x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right) + 0 \]
      rational_best-simplify-4 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)} \]
      rational_best-simplify-44 [=>] 0.4	\[ x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]

   if -4.99999999999999991e108 < x.im < 6.59999999999999948e92

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 0.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-2 [<=] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-47 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-44 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-2 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re} \]
      rational_best-simplify-48 [=>] 0.2	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 0.2

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

   if 6.59999999999999948e92 < x.im

   1. Initial program 33.1

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

   2. Simplified 33.3

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

      [Start] 33.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 33.1	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-2 [=>] 33.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-2 [<=] 33.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-47 [=>] 33.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-44 [=>] 33.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-2 [=>] 33.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re} \]
      rational_best-simplify-48 [=>] 33.3	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 33.3

      \[\leadsto \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0} \]

   4. Simplified 33.4

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

      [Start] 33.3	\[ x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right) + 0 \]
      rational_best-simplify-4 [=>] 33.3	\[ \color{blue}{x.re \cdot \left(\left(x.im \cdot x.im\right) \cdot -2 - \left(x.im \cdot x.im - x.re \cdot x.re\right)\right)} \]
      rational_best-simplify-46 [=>] 33.3	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(x.im \cdot x.im\right) \cdot -2 - x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-2 [=>] 33.3	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{-2 \cdot \left(x.im \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best-simplify-44 [=>] 33.3	\[ x.re \cdot \left(x.re \cdot x.re + \left(\color{blue}{x.im \cdot \left(-2 \cdot x.im\right)} - x.im \cdot x.im\right)\right) \]
      rational_best-simplify-48 [=>] 33.4	\[ x.re \cdot \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(-2 \cdot x.im - x.im\right)}\right) \]
      rational_best-simplify-2 [=>] 33.4	\[ x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(\color{blue}{x.im \cdot -2} - x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(-2 \cdot x.im - x.im\right) \cdot \left(x.re \cdot x.im\right) \]
      rational_best-simplify-2 [=>] 0.4	\[ \left(-2 \cdot x.im - x.im\right) \cdot \color{blue}{\left(x.im \cdot x.re\right)} \]
      rational_best-simplify-44 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(\left(-2 \cdot x.im - x.im\right) \cdot x.re\right)} \]

   7. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+108}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 6.6 \cdot 10^{+92}:\\ \;\;\;\;x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1088

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

Alternative 2
Error	0.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -4.3 \cdot 10^{+111}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1}\\ \end{array} \]

Alternative 3
Error	19.2
Cost	576

\[\frac{x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)}{1} \]

Alternative 4
Error	19.2
Cost	448

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

Alternative 5
Error	19.2
Cost	448

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

Alternative 6
Error	19.2
Cost	448

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

Reproduce

herbie shell --seed 2023092 
(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)))