math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.7%

Time: 11.3s

Precision: binary64

Cost: 7305

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• 45.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{+106} \lor \neg \left(x.im \leq 4 \cdot 10^{+101}\right):\\ \;\;\;\;\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

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 (or (<= x.im -3e+106) (not (<= x.im 4e+101)))
   (* (* x.im (+ x.im x.re)) (- x.re x.im))
   (- (* x.re (* 3.0 (* x.im x.re))) (pow x.im 3.0))))

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_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_im <= -3e+106) || !(x_46_im <= 4e+101)) {
		tmp = (x_46_im * (x_46_im + x_46_re)) * (x_46_re - x_46_im);
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - pow(x_46_im, 3.0);
	}
	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_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_46im <= (-3d+106)) .or. (.not. (x_46im <= 4d+101))) then
        tmp = (x_46im * (x_46im + x_46re)) * (x_46re - x_46im)
    else
        tmp = (x_46re * (3.0d0 * (x_46im * x_46re))) - (x_46im ** 3.0d0)
    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_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_im <= -3e+106) || !(x_46_im <= 4e+101)) {
		tmp = (x_46_im * (x_46_im + x_46_re)) * (x_46_re - x_46_im);
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - Math.pow(x_46_im, 3.0);
	}
	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_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_im <= -3e+106) or not (x_46_im <= 4e+101):
		tmp = (x_46_im * (x_46_im + x_46_re)) * (x_46_re - x_46_im)
	else:
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - math.pow(x_46_im, 3.0)
	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_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_im <= -3e+106) || !(x_46_im <= 4e+101))
		tmp = Float64(Float64(x_46_im * Float64(x_46_im + x_46_re)) * Float64(x_46_re - x_46_im));
	else
		tmp = Float64(Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re))) - (x_46_im ^ 3.0));
	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_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_im <= -3e+106) || ~((x_46_im <= 4e+101)))
		tmp = (x_46_im * (x_46_im + x_46_re)) * (x_46_re - x_46_im);
	else
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - (x_46_im ^ 3.0);
	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$46$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[Or[LessEqual[x$46$im, -3e+106], N[Not[LessEqual[x$46$im, 4e+101]], $MachinePrecision]], N[(N[(x$46$im * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	82.6%
Target	91.6%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if x.im < -3.0000000000000001e106 or 3.9999999999999999e101 < x.im

   1. Initial program 56.8%

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

   2. Applied egg-rr 66.7%

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

      [Start] 56.8	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      add-log-exp [=>] 56.8	\[ \color{blue}{\log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-un-lft-identity [=>] 56.8	\[ \log \color{blue}{\left(1 \cdot e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      log-prod [=>] 56.8	\[ \color{blue}{\left(\log 1 + \log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      metadata-eval [=>] 56.8	\[ \left(\color{blue}{0} + \log \left(e^{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right)\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      add-log-exp [<=] 56.8	\[ \left(0 + \color{blue}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 56.8	\[ \left(0 + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      difference-of-squares [=>] 66.7	\[ \left(0 + x.im \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      associate-*r* [=>] 66.7	\[ \left(0 + \color{blue}{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]

   3. Applied egg-rr 0.0%

      \[\leadsto \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \color{blue}{\left(0 + \frac{x.re \cdot 0}{0}\right)} \]
      Step-by-step derivation

      [Start] 66.7	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      add-log-exp [=>] 44.4	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \color{blue}{\log \left(e^{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right)} \]
      *-un-lft-identity [=>] 44.4	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \log \color{blue}{\left(1 \cdot e^{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right)} \]
      log-prod [=>] 44.4	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \color{blue}{\left(\log 1 + \log \left(e^{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right)\right)} \]
      metadata-eval [=>] 44.4	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(\color{blue}{0} + \log \left(e^{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right)\right) \]
      add-log-exp [<=] 66.7	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re}\right) \]
      *-commutative [=>] 66.7	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)}\right) \]
      *-commutative [=>] 66.7	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right)\right) \]
      flip-+ [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \color{blue}{\frac{\left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}{x.im \cdot x.re - x.im \cdot x.re}}\right) \]
      +-inverses [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \frac{\color{blue}{0}}{x.im \cdot x.re - x.im \cdot x.re}\right) \]
      +-inverses [<=] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \frac{\color{blue}{x.im \cdot x.re - x.im \cdot x.re}}{x.im \cdot x.re - x.im \cdot x.re}\right) \]
      *-commutative [<=] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \frac{\color{blue}{x.re \cdot x.im} - x.im \cdot x.re}{x.im \cdot x.re - x.im \cdot x.re}\right) \]
      *-commutative [<=] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + x.re \cdot \frac{x.re \cdot x.im - x.im \cdot x.re}{\color{blue}{x.re \cdot x.im} - x.im \cdot x.re}\right) \]
      associate-*r/ [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \color{blue}{\frac{x.re \cdot \left(x.re \cdot x.im - x.im \cdot x.re\right)}{x.re \cdot x.im - x.im \cdot x.re}}\right) \]
      *-commutative [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \frac{x.re \cdot \left(\color{blue}{x.im \cdot x.re} - x.im \cdot x.re\right)}{x.re \cdot x.im - x.im \cdot x.re}\right) \]
      +-inverses [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \frac{x.re \cdot \color{blue}{0}}{x.re \cdot x.im - x.im \cdot x.re}\right) \]
      *-commutative [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \frac{x.re \cdot 0}{\color{blue}{x.im \cdot x.re} - x.im \cdot x.re}\right) \]
      +-inverses [=>] 0.0	\[ \left(0 + \left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\right) + \left(0 + \frac{x.re \cdot 0}{\color{blue}{0}}\right) \]

   4. Simplified 100.0%

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

   if -3.0000000000000001e106 < x.im < 3.9999999999999999e101

   1. Initial program 91.7%

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

   2. Simplified 99.8%

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

      [Start] 91.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      +-commutative [=>] 91.7	\[ \color{blue}{\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im} \]
      *-commutative [=>] 91.7	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} \]
      sub-neg [=>] 91.7	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} \]
      distribute-lft-in [=>] 91.7	\[ \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right) + x.im \cdot \left(-x.im \cdot x.im\right)\right)} \]
      associate-+r+ [=>] 91.7	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + x.im \cdot \left(-x.im \cdot x.im\right)} \]
      distribute-rgt-neg-out [=>] 91.7	\[ \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) + \color{blue}{\left(-x.im \cdot \left(x.im \cdot x.im\right)\right)} \]
      unsub-neg [=>] 91.7	\[ \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + x.im \cdot \left(x.re \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right)} \]
      associate-*r* [=>] 99.7	\[ \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re + \color{blue}{\left(x.im \cdot x.re\right) \cdot x.re}\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
      distribute-rgt-out [=>] 99.8	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.im + x.im \cdot x.re\right) + x.im \cdot x.re\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
      *-commutative [=>] 99.8	\[ x.re \cdot \left(\left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) + x.im \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
      count-2 [=>] 99.8	\[ x.re \cdot \left(\color{blue}{2 \cdot \left(x.im \cdot x.re\right)} + x.im \cdot x.re\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
      distribute-lft1-in [=>] 99.8	\[ x.re \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \left(x.im \cdot x.re\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
      metadata-eval [=>] 99.8	\[ x.re \cdot \left(\color{blue}{3} \cdot \left(x.im \cdot x.re\right)\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
      *-commutative [=>] 99.8	\[ x.re \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot 3\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
      *-commutative [<=] 99.8	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.im\right)} \cdot 3\right) - x.im \cdot \left(x.im \cdot x.im\right) \]
      associate-*r* [<=] 99.7	\[ x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} - x.im \cdot \left(x.im \cdot x.im\right) \]
      cube-unmult [=>] 99.8	\[ x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - \color{blue}{{x.im}^{3}} \]

   3. Taylor expanded in x.re around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{+106} \lor \neg \left(x.im \leq 4 \cdot 10^{+101}\right):\\ \;\;\;\;\left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	2500

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

Alternative 2
Accuracy	98.2%
Cost	1744

\[\begin{array}{l} t_0 := \left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\\ t_1 := x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -7.5 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 1.5 \cdot 10^{-117}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 7200000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	1480

\[\begin{array}{l} t_0 := \left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\\ \mathbf{if}\;x.im \leq -2 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7200000000000:\\ \;\;\;\;t_0 + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\right) \cdot \left(\left(x.im + x.re\right) \cdot \frac{1}{x.im + x.re}\right)\\ \end{array} \]

Alternative 4
Accuracy	95.9%
Cost	1232

\[\begin{array}{l} t_0 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ t_1 := x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -9.4 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-101}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	98.4%
Cost	1232

\[\begin{array}{l} t_0 := \left(x.im \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right)\\ t_1 := x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -1.7 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 7 \cdot 10^{-108}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 2000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	89.5%
Cost	1104

\[\begin{array}{l} t_0 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ t_1 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.1 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq -6 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{-32}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 6.6 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	75.6%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.15 \cdot 10^{+89} \lor \neg \left(x.re \leq -4.7 \cdot 10^{+20} \lor \neg \left(x.re \leq -1.12 \cdot 10^{-55}\right) \land x.re \leq 4.9 \cdot 10^{-17}\right):\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 8
Accuracy	81.6%
Cost	976

\[\begin{array}{l} t_0 := 3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ t_1 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.05 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -1.26 \cdot 10^{-55}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	81.6%
Cost	976

\[\begin{array}{l} t_0 := 3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ t_1 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -1.95 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -7.2 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -1.26 \cdot 10^{-55}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.9 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	81.6%
Cost	976

\[\begin{array}{l} t_0 := \left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -1.26 \cdot 10^{-55}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	71.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.35 \cdot 10^{+179} \lor \neg \left(x.re \leq 5.2 \cdot 10^{+138}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 12
Accuracy	34.9%
Cost	320

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

Alternative 13
Accuracy	2.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023159 
(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)))