math.cube on complex, imaginary part

Average Error: 7.9 → 0.2

Time: 9.8s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+133}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+
  (* (- (* x.re x.re) (* x.im x.im)) x.im)
  (* (+ (* x.re x.im) (* x.im x.re)) x.re)))

↓

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re -1e+140)
   (* x.re (* x.re (* 3.0 x.im)))
   (if (<= x.re 2e+133)
     (+
      (* x.re (+ (* x.re x.im) (* x.re x.im)))
      (* x.im (- (* x.re x.re) (* x.im x.im))))
     (* (* 3.0 x.re) (* 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im));
	} else if (x_46_re <= 2e+133) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	} else {
		tmp = (3.0 * x_46_re) * (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_46im) + (((x_46re * x_46im) + (x_46im * x_46re)) * x_46re)
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_46re <= (-1d+140)) then
        tmp = x_46re * (x_46re * (3.0d0 * x_46im))
    else if (x_46re <= 2d+133) then
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + (x_46im * ((x_46re * x_46re) - (x_46im * x_46im)))
    else
        tmp = (3.0d0 * x_46re) * (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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
}

↓

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= -1e+140) {
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im));
	} else if (x_46_re <= 2e+133) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	} else {
		tmp = (3.0 * x_46_re) * (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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re)

↓

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= -1e+140:
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im))
	elif x_46_re <= 2e+133:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)))
	else:
		tmp = (3.0 * x_46_re) * (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_im) + Float64(Float64(Float64(x_46_re * x_46_im) + Float64(x_46_im * x_46_re)) * x_46_re))
end

↓

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= -1e+140)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(3.0 * x_46_im)));
	elseif (x_46_re <= 2e+133)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))));
	else
		tmp = Float64(Float64(3.0 * x_46_re) * 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_im) + (((x_46_re * x_46_im) + (x_46_im * x_46_re)) * x_46_re);
end

↓

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= -1e+140)
		tmp = x_46_re * (x_46_re * (3.0 * x_46_im));
	elseif (x_46_re <= 2e+133)
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	else
		tmp = (3.0 * x_46_re) * (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$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, -1e+140], N[(x$46$re * N[(x$46$re * N[(3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2e+133], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * x$46$re), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.9
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x.re < -1.00000000000000006e140

   1. Initial program 55.1

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

   2. Simplified 55.5

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}} \]
      Proof
      (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 x.re 3))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 3 x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 x.re) x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (+.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.re) (+.f64 (+.f64 x.re x.re) x.re))) (pow.f64 x.im 3)): 14 points increase in error, 43 points decrease in error
      (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (+.f64 x.re x.re) x.re) (*.f64 x.im x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (Rewrite=> count-2_binary64 (*.f64 2 x.re)) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 2 1) x.re)) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite=> metadata-eval 3) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re)))) (pow.f64 x.im 3)): 24 points increase in error, 19 points decrease in error
      (-.f64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.im x.im) x.im))): 10 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 3 (*.f64 x.re (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)))) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 3 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) x.im))) (*.f64 (*.f64 x.im x.im) x.im)): 43 points increase in error, 5 points decrease in error
      (-.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 3 (*.f64 x.re x.re)) x.im)) (*.f64 (*.f64 x.im x.im) x.im)): 11 points increase in error, 21 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (*.f64 x.re x.re)) (*.f64 x.re x.re))) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.re) x.re)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x.im (-.f64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (*.f64 x.re x.re)) (*.f64 x.im x.im)))): 0 points increase in error, 1 points decrease in error
      (*.f64 x.im (Rewrite<= associate-+r-_binary64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) (*.f64 (+.f64 x.re x.re) x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.im (*.f64 (+.f64 x.re x.re) x.re)))): 19 points increase in error, 11 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im (+.f64 x.re x.re)) x.re))): 2 points increase in error, 11 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.re x.im) (*.f64 x.re x.im))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (*.f64 x.re x.im) (Rewrite<= *-commutative_binary64 (*.f64 x.im x.re))) x.re)): 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.im)) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.8

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

   4. Taylor expanded in x.re around inf 55.2

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

   5. Simplified 0.4

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)} \]
      Proof
      (*.f64 x.re (*.f64 x.re (*.f64 x.im 3))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (*.f64 x.re (*.f64 x.im (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 3) (sqrt.f64 3)))))): 61 points increase in error, 41 points decrease in error
      (*.f64 x.re (*.f64 x.re (*.f64 x.im (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 3) 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sqrt.f64 3) 2) x.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) (*.f64 (pow.f64 (sqrt.f64 3) 2) x.im))): 56 points increase in error, 27 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)) (*.f64 (pow.f64 (sqrt.f64 3) 2) x.im)): 0 points increase in error, 0 points decrease in error

   if -1.00000000000000006e140 < x.re < 2e133

   1. Initial program 0.2

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

   if 2e133 < x.re

   1. Initial program 52.6

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

   2. Simplified 52.7

      \[\leadsto \color{blue}{x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}} \]
      Proof
      (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 x.re 3))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= *-commutative_binary64 (*.f64 3 x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 x.re) x.re)))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 x.im (*.f64 x.re (+.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.re) (+.f64 (+.f64 x.re x.re) x.re))) (pow.f64 x.im 3)): 14 points increase in error, 43 points decrease in error
      (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (+.f64 x.re x.re) x.re) (*.f64 x.im x.re))) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (Rewrite=> count-2_binary64 (*.f64 2 x.re)) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 2 1) x.re)) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite=> metadata-eval 3) x.re) (*.f64 x.im x.re)) (pow.f64 x.im 3)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re)))) (pow.f64 x.im 3)): 24 points increase in error, 19 points decrease in error
      (-.f64 (*.f64 3 (*.f64 x.re (*.f64 x.im x.re))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x.im x.im) x.im))): 10 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 3 (*.f64 x.re (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)))) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 3 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) x.im))) (*.f64 (*.f64 x.im x.im) x.im)): 43 points increase in error, 5 points decrease in error
      (-.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 3 (*.f64 x.re x.re)) x.im)) (*.f64 (*.f64 x.im x.im) x.im)): 11 points increase in error, 21 points decrease in error
      (-.f64 (*.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 2 1)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 2 (*.f64 x.re x.re)) (*.f64 x.re x.re))) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 x.re) x.re)) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (+.f64 (*.f64 (Rewrite<= count-2_binary64 (+.f64 x.re x.re)) x.re) (*.f64 x.re x.re)) x.im) (*.f64 (*.f64 x.im x.im) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x.im (-.f64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (*.f64 x.re x.re)) (*.f64 x.im x.im)))): 0 points increase in error, 1 points decrease in error
      (*.f64 x.im (Rewrite<= associate-+r-_binary64 (+.f64 (*.f64 (+.f64 x.re x.re) x.re) (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.im (Rewrite=> +-commutative_binary64 (+.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) (*.f64 (+.f64 x.re x.re) x.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 x.im (*.f64 (+.f64 x.re x.re) x.re)))): 19 points increase in error, 11 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im (+.f64 x.re x.re)) x.re))): 2 points increase in error, 11 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 x.re x.im) (*.f64 x.re x.im))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x.im (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im))) (*.f64 (+.f64 (*.f64 x.re x.im) (Rewrite<= *-commutative_binary64 (*.f64 x.im x.re))) x.re)): 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.im)) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x.im around 0 52.6

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

   4. Simplified 0.4

      \[\leadsto \color{blue}{x.re \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)} - {x.im}^{3} \]
      Proof
      (*.f64 x.re (*.f64 x.im (*.f64 3 x.re))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (*.f64 x.im (Rewrite<= *-commutative_binary64 (*.f64 x.re 3)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x.re (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.im x.re) 3))): 21 points increase in error, 30 points decrease in error
      (*.f64 x.re (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re x.im)) 3)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re (*.f64 x.re x.im)) 3)): 27 points increase in error, 31 points decrease in error
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x.re x.re) x.im)) 3): 54 points increase in error, 10 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x.re 2)) x.im) 3): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 (pow.f64 x.re 2) x.im))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.3

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

   6. Taylor expanded in x.re around inf 52.6

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

   7. Simplified 0.4

      \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(x.re \cdot 3\right)} \]
      Proof
      (*.f64 (*.f64 x.im x.re) (*.f64 x.re 3)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 x.im x.re) x.re) 3)): 27 points increase in error, 30 points decrease in error
      (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 x.im (*.f64 x.re x.re))) 3): 54 points increase in error, 10 points decrease in error
      (*.f64 (*.f64 x.im (Rewrite<= unpow2_binary64 (pow.f64 x.re 2))) 3): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 x.re 2) x.im)) 3): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 (pow.f64 x.re 2) x.im))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{+140}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+133}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7040

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

Alternative 2
Error	0.2
Cost	7040

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

Alternative 3
Error	0.2
Cost	1216

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

Alternative 4
Error	0.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -3.0041067103933783 \cdot 10^{+119}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.9833399147460208 \cdot 10^{+81}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 5
Error	7.1
Cost	712

\[\begin{array}{l} t_0 := \left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -2.206416045642465 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 5.763179562958071 \cdot 10^{+31}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	7.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.206416045642465 \cdot 10^{-18}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 5.763179562958071 \cdot 10^{+31}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot x.re\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 7
Error	28.6
Cost	384

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

Reproduce

herbie shell --seed 2022296 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))