math.cube on complex, real part

Average Error: 7.3 → 0.4

Time: 15.8s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right) - x.re \cdot x.re\right)\\ \end{array} \]

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 -1e+156)
   (- (* (+ x.re x.im) (* x.re (- x.im))) (* x.im (* x.im (+ x.re x.re))))
   (if (<= x.im 1.65e+70)
     (-
      (* 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)) (* x.re x.re))))))

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 <= -1e+156) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * -x_46_im)) - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	} else if (x_46_im <= 1.65e+70) {
		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)) - (x_46_re * x_46_re));
	}
	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 <= (-1d+156)) then
        tmp = ((x_46re + x_46im) * (x_46re * -x_46im)) - (x_46im * (x_46im * (x_46re + x_46re)))
    else if (x_46im <= 1.65d+70) 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))) - (x_46re * x_46re))
    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 <= -1e+156) {
		tmp = ((x_46_re + x_46_im) * (x_46_re * -x_46_im)) - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	} else if (x_46_im <= 1.65e+70) {
		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)) - (x_46_re * x_46_re));
	}
	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 <= -1e+156:
		tmp = ((x_46_re + x_46_im) * (x_46_re * -x_46_im)) - (x_46_im * (x_46_im * (x_46_re + x_46_re)))
	elif x_46_im <= 1.65e+70:
		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)) - (x_46_re * x_46_re))
	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 <= -1e+156)
		tmp = Float64(Float64(Float64(x_46_re + x_46_im) * Float64(x_46_re * Float64(-x_46_im))) - Float64(x_46_im * Float64(x_46_im * Float64(x_46_re + x_46_re))));
	elseif (x_46_im <= 1.65e+70)
		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(Float64(x_46_im * Float64(x_46_re * -3.0)) - Float64(x_46_re * x_46_re)));
	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 <= -1e+156)
		tmp = ((x_46_re + x_46_im) * (x_46_re * -x_46_im)) - (x_46_im * (x_46_im * (x_46_re + x_46_re)));
	elseif (x_46_im <= 1.65e+70)
		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)) - (x_46_re * x_46_re));
	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, -1e+156], 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 * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.65e+70], 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[(N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.3
Target	0.3
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x.im < -9.9999999999999998e155

   1. Initial program 64.0

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

   2. Simplified 0.4

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

      [Start] 64.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-34 [=>] 64.0	\[ x.re \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 64.0	\[ x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best_45_simplify-14 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \]

   3. Taylor expanded in x.re around 0 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-1 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(-1 \cdot x.im\right)\right)} - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-18 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot -1\right)}\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-62 [<=] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]

   if -9.9999999999999998e155 < x.im < 1.65000000000000008e70

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 0.3	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 0.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best_45_simplify-18 [=>] 0.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational_best_45_simplify-18 [<=] 0.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im \cdot x.re + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best_45_simplify-14 [=>] 0.3	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   if 1.65000000000000008e70 < x.im

   1. Initial program 27.0

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

   2. Simplified 0.4

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

      [Start] 27.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 27.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-34 [=>] 27.0	\[ x.re \cdot \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 27.0	\[ x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_45_simplify-18 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best_45_simplify-14 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \left(x.re - x.im\right)\right) - x.im \cdot \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \]

   3. Taylor expanded in x.re around 0 0.9

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

   4. Simplified 0.9

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

      [Start] 0.9	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-1 [=>] 0.9	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(-1 \cdot x.im\right)\right)} - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-18 [=>] 0.9	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot -1\right)}\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]
      rational_best_45_simplify-62 [<=] 0.9	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right) \]

   5. Applied egg-rr 0.9

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

   6. Simplified 0.9

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

      [Start] 0.9	\[ \left(-x.re \cdot x.re\right) \cdot x.im + x.im \cdot \left(x.im \cdot \left(\left(-x.re\right) - \left(x.re + x.re\right)\right)\right) \]
      rational_best_45_simplify-14 [=>] 0.9	\[ \color{blue}{x.im \cdot \left(\left(-x.re \cdot x.re\right) + x.im \cdot \left(\left(-x.re\right) - \left(x.re + x.re\right)\right)\right)} \]

   7. Applied egg-rr 27.5

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

   8. Simplified 0.9

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

      [Start] 27.5	\[ \left(-x.re\right) \cdot \left(x.im \cdot x.im\right) + x.im \cdot \left(x.im \cdot \left(x.re \cdot -2\right) - x.re \cdot x.re\right) \]
      rational_best_45_simplify-9 [=>] 27.5	\[ \left(-x.re\right) \cdot \left(x.im \cdot x.im\right) + \color{blue}{\left(-x.im\right) \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.re \cdot -2\right)\right)} \]
      rational_best_45_simplify-13 [=>] 27.5	\[ \left(-x.re\right) \cdot \left(x.im \cdot x.im\right) + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(-x.im\right) - \left(-x.im\right) \cdot \left(x.im \cdot \left(x.re \cdot -2\right)\right)\right)} \]
      rational_best_45_simplify-109 [=>] 27.5	\[ \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot \left(-x.im\right) + \left(-x.re\right) \cdot \left(x.im \cdot x.im\right)\right) - \left(-x.im\right) \cdot \left(x.im \cdot \left(x.re \cdot -2\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+156}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right) - x.re \cdot x.re\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	2368

\[\begin{array}{l} t_0 := x.re \cdot \left(\left(x.re - x.im\right) \cdot \frac{x.re + x.im}{x.re + x.im}\right)\\ x.re \cdot t_0 + x.im \cdot \left(t_0 - x.re \cdot \left(x.im + x.im\right)\right) \end{array} \]

Alternative 2
Error	0.3
Cost	1224

\[\begin{array}{l} t_0 := x.im \cdot \left(\left(-x.re \cdot x.re\right) + -3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -6.5 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.im \leq -7 \cdot 10^{+104}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.im \cdot \left(x.re + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(-x.re \cdot x.re\right) + -3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 4
Error	5.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.9 \cdot 10^{-9}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right) - x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.re - \frac{x.im \cdot \left(x.im + x.im\right)}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(-x.re \cdot x.re\right) + -3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 5
Error	0.2
Cost	1088

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

Alternative 6
Error	5.4
Cost	1032

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right) - x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 3.9 \cdot 10^{-55}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(-x.re \cdot x.re\right) + -3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 7
Error	5.4
Cost	968

\[\begin{array}{l} t_0 := x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right) - x.re \cdot x.re\right)\\ \mathbf{if}\;x.im \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	5.4
Cost	840

\[\begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im - x.re\right)\right)\\ \mathbf{if}\;x.im \leq -3.3 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	5.3
Cost	712

\[\begin{array}{l} t_0 := \left(x.re \cdot x.im\right) \cdot \left(x.im \cdot -3\right)\\ \mathbf{if}\;x.im \leq -4.6 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{-56}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	28.0
Cost	320

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

Reproduce

herbie shell --seed 2023102 
(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)))