math.cube on complex, real part

Average Error: 7.4 → 0.2

Time: 10.5s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 6.128515027557732 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

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+150)
   (* x.im (* -3.0 (* x.re x.im)))
   (if (<= x.im 6.128515027557732e+90)
     (+
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* (* x.re x.im) -2.0)))
     (* -3.0 (* x.im (* x.re x.im))))))

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+150) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_im <= 6.128515027557732e+90) {
		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) * -2.0));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	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+150)) then
        tmp = x_46im * ((-3.0d0) * (x_46re * x_46im))
    else if (x_46im <= 6.128515027557732d+90) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46im * ((x_46re * x_46im) * (-2.0d0)))
    else
        tmp = (-3.0d0) * (x_46im * (x_46re * x_46im))
    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+150) {
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	} else if (x_46_im <= 6.128515027557732e+90) {
		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) * -2.0));
	} else {
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	}
	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+150:
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im))
	elif x_46_im <= 6.128515027557732e+90:
		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) * -2.0))
	else:
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im))
	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+150)
		tmp = Float64(x_46_im * Float64(-3.0 * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 6.128515027557732e+90)
		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(Float64(x_46_re * x_46_im) * -2.0)));
	else
		tmp = Float64(-3.0 * Float64(x_46_im * Float64(x_46_re * x_46_im)));
	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+150)
		tmp = x_46_im * (-3.0 * (x_46_re * x_46_im));
	elseif (x_46_im <= 6.128515027557732e+90)
		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) * -2.0));
	else
		tmp = -3.0 * (x_46_im * (x_46_re * x_46_im));
	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+150], N[(x$46$im * N[(-3.0 * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6.128515027557732e+90], 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] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-3.0 * N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.4
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x.im < -9.99999999999999981e149

   1. Initial program 60.9

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

   2. Simplified 60.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)} \]
      Proof
      (fma.f64 x.re (*.f64 x.im (*.f64 x.im -3)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (*.f64 x.im (*.f64 x.im (Rewrite<= metadata-eval (-.f64 -1 2)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.im) (-.f64 -1 2))) (pow.f64 x.re 3)): 17 points increase in error, 8 points decrease in error
      (fma.f64 x.re (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 (*.f64 x.im x.im)) (*.f64 2 (*.f64 x.im x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 x.im x.im))) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x.im) x.im)) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.im) x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.im x.im)) x.im)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (neg.f64 (*.f64 (+.f64 x.im x.im) x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.re x.re) x.re))): 16 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (*.f64 (*.f64 x.re x.re) x.re))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (Rewrite=> associate-*l*_binary64 (*.f64 x.re (*.f64 x.re x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-lft-out_binary64 (*.f64 x.re (+.f64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (*.f64 x.re x.re)))): 1 points increase in error, 2 points decrease in error
      (*.f64 x.re (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x.re x.re) (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x.re x.re) (*.f64 (neg.f64 x.im) x.im)) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 1 points increase in error, 1 points decrease in error
      (*.f64 x.re (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.re (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 12 points increase in error, 25 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re (+.f64 x.im x.im)) (neg.f64 x.im)))): 4 points increase in error, 15 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.im x.re) (*.f64 x.im x.re))) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)) (*.f64 x.im x.re)) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re)) (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.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))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x.re around 0 60.9

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)} \]
      Proof
      (*.f64 x.im (*.f64 -3 (*.f64 x.re x.im))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -3 x.re) x.im))): 31 points increase in error, 27 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 -3 x.re) x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -3 x.re) (*.f64 x.im x.im))): 56 points increase in error, 28 points decrease in error
      (*.f64 (*.f64 -3 x.re) (Rewrite<= unpow2_binary64 (pow.f64 x.im 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -3 (*.f64 x.re (pow.f64 x.im 2)))): 21 points increase in error, 28 points decrease in error

   if -9.99999999999999981e149 < x.im < 6.12851502755773231e90

   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. Applied egg-rr 0.2

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

   if 6.12851502755773231e90 < 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 33.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.im \cdot \left(x.im \cdot -3\right), {x.re}^{3}\right)} \]
      Proof
      (fma.f64 x.re (*.f64 x.im (*.f64 x.im -3)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (*.f64 x.im (*.f64 x.im (Rewrite<= metadata-eval (-.f64 -1 2)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.im) (-.f64 -1 2))) (pow.f64 x.re 3)): 17 points increase in error, 8 points decrease in error
      (fma.f64 x.re (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 -1 (*.f64 x.im x.im)) (*.f64 2 (*.f64 x.im x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (*.f64 x.im x.im))) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 x.im) x.im)) (*.f64 2 (*.f64 x.im x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.im) x.im))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (-.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.im x.im)) x.im)) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (neg.f64 (*.f64 (+.f64 x.im x.im) x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (pow.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.re x.re) x.re))): 16 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (*.f64 (*.f64 x.re x.re) x.re))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))) (Rewrite=> associate-*l*_binary64 (*.f64 x.re (*.f64 x.re x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-lft-out_binary64 (*.f64 x.re (+.f64 (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))) (*.f64 x.re x.re)))): 1 points increase in error, 2 points decrease in error
      (*.f64 x.re (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x.re x.re) (+.f64 (*.f64 (neg.f64 x.im) x.im) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 x.re x.re) (*.f64 (neg.f64 x.im) x.im)) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 1 points increase in error, 1 points decrease in error
      (*.f64 x.re (+.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 x.im x.im) (neg.f64 x.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.re (*.f64 (+.f64 x.im x.im) (neg.f64 x.im))))): 12 points increase in error, 25 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re (+.f64 x.im x.im)) (neg.f64 x.im)))): 4 points increase in error, 15 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.im x.re) (*.f64 x.im x.re))) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)) (*.f64 x.im x.re)) (neg.f64 x.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.re (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.re)) (neg.f64 (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.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))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x.re around 0 33.4

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

   4. Simplified 0.5

      \[\leadsto \color{blue}{x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)} \]
      Proof
      (*.f64 x.im (*.f64 -3 (*.f64 x.re x.im))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -3 x.re) x.im))): 31 points increase in error, 27 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 -3 x.re) x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -3 x.re) (*.f64 x.im x.im))): 56 points increase in error, 28 points decrease in error
      (*.f64 (*.f64 -3 x.re) (Rewrite<= unpow2_binary64 (pow.f64 x.im 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -3 (*.f64 x.re (pow.f64 x.im 2)))): 21 points increase in error, 28 points decrease in error

   5. Applied egg-rr 1.5

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

   6. Applied egg-rr 0.6

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+150}:\\ \;\;\;\;x.im \cdot \left(-3 \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 6.128515027557732 \cdot 10^{+90}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7040

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

Alternative 2
Error	1.0
Cost	968

\[\begin{array}{l} t_0 := -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -12753689248353060:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 4.027667113252067 \cdot 10^{+44}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + -3 \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	6.0
Cost	712

\[\begin{array}{l} t_0 := -3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{if}\;x.im \leq -6.511203333667252 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 6.242000797767714 \cdot 10^{-81}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	28.0
Cost	320

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

Reproduce

herbie shell --seed 2022317 
(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)))