math.cube on complex, imaginary part

Average Error: 88.1% → 99.7%

Time: 18.1s

Precision: binary64

Cost: 7304.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4 \cdot 10^{+109}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 9 \cdot 10^{+84}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

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 -4e+109)
   (* 3.0 (* x.re (* x.re x.im)))
   (if (<= x.re 9e+84)
     (- (* 3.0 (* x.im (* x.re x.re))) (pow x.im 3.0))
     (* x.re (* x.im (* x.re 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_re <= -4e+109) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 9e+84) {
		tmp = (3.0 * (x_46_im * (x_46_re * x_46_re))) - pow(x_46_im, 3.0);
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 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_46re <= (-4d+109)) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46re <= 9d+84) then
        tmp = (3.0d0 * (x_46im * (x_46re * x_46re))) - (x_46im ** 3.0d0)
    else
        tmp = x_46re * (x_46im * (x_46re * 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_re <= -4e+109) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_re <= 9e+84) {
		tmp = (3.0 * (x_46_im * (x_46_re * x_46_re))) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = x_46_re * (x_46_im * (x_46_re * 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_re <= -4e+109:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_re <= 9e+84:
		tmp = (3.0 * (x_46_im * (x_46_re * x_46_re))) - math.pow(x_46_im, 3.0)
	else:
		tmp = x_46_re * (x_46_im * (x_46_re * 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_re <= -4e+109)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_re <= 9e+84)
		tmp = Float64(Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re))) - (x_46_im ^ 3.0));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 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_re <= -4e+109)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_re <= 9e+84)
		tmp = (3.0 * (x_46_im * (x_46_re * x_46_re))) - (x_46_im ^ 3.0);
	else
		tmp = x_46_re * (x_46_im * (x_46_re * 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[LessEqual[x$46$re, -4e+109], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 9e+84], N[(N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.1%
Target	99.6%
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 3 regimes

2. if x.re < -3.99999999999999993e109

   1. Initial program 36.9

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

   2. Simplified 36.6

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

      [Start] 36.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 36.9	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [<=] 36.9	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.re \]
      distribute-rgt-out [=>] 36.9	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \cdot x.re \]
      associate-*l* [=>] 36.6	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{x.im \cdot \left(\left(x.re + x.re\right) \cdot x.re\right)} \]
      distribute-lft-out [=>] 36.6	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right) \cdot x.re\right)} \]
      +-commutative [<=] 36.6	\[ x.im \cdot \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
      associate-+r- [=>] 36.6	\[ x.im \cdot \color{blue}{\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]
      count-2 [=>] 36.6	\[ x.im \cdot \left(\left(\color{blue}{\left(2 \cdot x.re\right)} \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right) \]
      associate-*l* [=>] 36.6	\[ x.im \cdot \left(\left(\color{blue}{2 \cdot \left(x.re \cdot x.re\right)} + x.re \cdot x.re\right) - x.im \cdot x.im\right) \]
      distribute-lft1-in [=>] 36.6	\[ x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]
      fma-neg [=>] 36.6	\[ x.im \cdot \color{blue}{\mathsf{fma}\left(2 + 1, x.re \cdot x.re, -x.im \cdot x.im\right)} \]
      metadata-eval [=>] 36.6	\[ x.im \cdot \mathsf{fma}\left(\color{blue}{3}, x.re \cdot x.re, -x.im \cdot x.im\right) \]

   3. Taylor expanded in x.im around 0 36.9

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

   4. Simplified 99.4

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

      [Start] 36.9	\[ 3 \cdot \left({x.re}^{2} \cdot x.im\right) \]
      unpow2 [=>] 36.9	\[ 3 \cdot \left(\color{blue}{\left(x.re \cdot x.re\right)} \cdot x.im\right) \]
      associate-*l* [=>] 99.4	\[ 3 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]

   if -3.99999999999999993e109 < x.re < 8.9999999999999994e84

   1. Initial program 99.7

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

   2. Simplified 99.8

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

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

   if 8.9999999999999994e84 < x.re

   1. Initial program 51.3

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

   2. Simplified 51.0

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

      [Start] 51.3	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [=>] 51.3	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      *-commutative [<=] 51.3	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.re \]
      distribute-rgt-out [=>] 51.3	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{\left(x.im \cdot \left(x.re + x.re\right)\right)} \cdot x.re \]
      associate-*l* [=>] 51.0	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{x.im \cdot \left(\left(x.re + x.re\right) \cdot x.re\right)} \]
      distribute-lft-out [=>] 50.9	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right) \cdot x.re\right)} \]
      +-commutative [<=] 50.9	\[ x.im \cdot \color{blue}{\left(\left(x.re + x.re\right) \cdot x.re + \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
      associate-+r- [=>] 50.9	\[ x.im \cdot \color{blue}{\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right)} \]
      count-2 [=>] 50.9	\[ x.im \cdot \left(\left(\color{blue}{\left(2 \cdot x.re\right)} \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right) \]
      associate-*l* [=>] 50.9	\[ x.im \cdot \left(\left(\color{blue}{2 \cdot \left(x.re \cdot x.re\right)} + x.re \cdot x.re\right) - x.im \cdot x.im\right) \]
      distribute-lft1-in [=>] 50.9	\[ x.im \cdot \left(\color{blue}{\left(2 + 1\right) \cdot \left(x.re \cdot x.re\right)} - x.im \cdot x.im\right) \]
      fma-neg [=>] 51.0	\[ x.im \cdot \color{blue}{\mathsf{fma}\left(2 + 1, x.re \cdot x.re, -x.im \cdot x.im\right)} \]
      metadata-eval [=>] 51.0	\[ x.im \cdot \mathsf{fma}\left(\color{blue}{3}, x.re \cdot x.re, -x.im \cdot x.im\right) \]

   3. Taylor expanded in x.im around 0 50.8

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

   4. Simplified 50.8

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

      [Start] 50.8	\[ 3 \cdot \left({x.re}^{2} \cdot x.im\right) \]
      *-commutative [=>] 50.8	\[ 3 \cdot \color{blue}{\left(x.im \cdot {x.re}^{2}\right)} \]
      unpow2 [=>] 50.8	\[ 3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \]

   5. Taylor expanded in x.im around 0 50.8

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

   6. Simplified 99.0

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

      [Start] 50.8	\[ 3 \cdot \left({x.re}^{2} \cdot x.im\right) \]
      associate-*r* [=>] 50.5	\[ \color{blue}{\left(3 \cdot {x.re}^{2}\right) \cdot x.im} \]
      unpow2 [=>] 50.5	\[ \left(3 \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) \cdot x.im \]
      *-commutative [<=] 50.5	\[ \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right)} \cdot x.im \]
      associate-*l* [=>] 50.5	\[ \color{blue}{\left(x.re \cdot \left(x.re \cdot 3\right)\right)} \cdot x.im \]
      associate-*l* [=>] 99.0	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot 3\right) \cdot x.im\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4 \cdot 10^{+109}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 9 \cdot 10^{+84}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	99.7%
Cost	7040.00

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

Alternative 2
Error	99.7%
Cost	7040.00

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

Alternative 3
Error	99.6%
Cost	1352.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{+154}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 4
Error	99.6%
Cost	968.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.96 \cdot 10^{+96}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{+96}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 5
Error	91.7%
Cost	840.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -4.1 \cdot 10^{-23}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.55 \cdot 10^{-82}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 6
Error	91.7%
Cost	840.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.85 \cdot 10^{-25}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.55 \cdot 10^{-82}:\\ \;\;\;\;\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 7
Error	80.0%
Cost	713.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -2.9 \cdot 10^{-26} \lor \neg \left(x.re \leq 2.55 \cdot 10^{-82}\right):\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 8
Error	91.5%
Cost	713.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{-27} \lor \neg \left(x.re \leq 2.55 \cdot 10^{-82}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 9
Error	91.5%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{-27}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.55 \cdot 10^{-82}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 10
Error	91.5%
Cost	712.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -7 \cdot 10^{-27}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.re \leq 2.55 \cdot 10^{-82}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right)\\ \end{array} \]

Alternative 11
Error	59.6%
Cost	649.00

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1750000000000 \lor \neg \left(x.re \leq 8.8 \cdot 10^{+23}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.im\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 12
Error	31.0%
Cost	320.00

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

Alternative 13
Error	33.0%
Cost	320.00

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

Alternative 14
Error	26.7%
Cost	192.00

\[x.im \cdot 0 \]

Reproduce

herbie shell --seed 2023093 
(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)))