math.cube on complex, real part

Average Error: 6.9 → 0.2

Time: 11.5s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 10^{+133}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(-x.im\right)\right) \cdot \left(x.im \cdot 3\right)\\ \end{array} \]

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 -8e+152)
   (* x.im (* (* x.im x.re) -3.0))
   (if (<= x.im 1e+133)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* x.re (+ x.im x.im))))
     (* (* x.re (- x.im)) (* 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_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 <= -8e+152) {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	} else if (x_46_im <= 1e+133) {
		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_re * -x_46_im) * (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_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 <= (-8d+152)) then
        tmp = x_46im * ((x_46im * x_46re) * (-3.0d0))
    else if (x_46im <= 1d+133) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * (x_46re * (x_46im + x_46im)))
    else
        tmp = (x_46re * -x_46im) * (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_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 <= -8e+152) {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	} else if (x_46_im <= 1e+133) {
		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_re * -x_46_im) * (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_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 <= -8e+152:
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0)
	elif x_46_im <= 1e+133:
		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_re * -x_46_im) * (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_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 <= -8e+152)
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) * -3.0));
	elseif (x_46_im <= 1e+133)
		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(Float64(x_46_re * Float64(-x_46_im)) * Float64(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_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 <= -8e+152)
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	elseif (x_46_im <= 1e+133)
		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_re * -x_46_im) * (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$46$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, -8e+152], N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1e+133], 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[(N[(x$46$re * (-x$46$im)), $MachinePrecision] * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	6.9
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 < -8.0000000000000004e152

   1. Initial program 62.2

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

   2. Simplified 62.2

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

      [Start] 62.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 62.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 62.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 62.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 62.2

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

   4. Simplified 0.4

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

      [Start] 62.2	\[ \left(x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + 0\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-4 [=>] 62.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-34 [=>] 62.2	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -1\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-9 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   7. Applied egg-rr 0.4

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

   8. Taylor expanded in x.re around 0 0.4

      \[\leadsto \color{blue}{\left(-1 \cdot x.im - 2 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

   9. Simplified 0.4

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

      [Start] 0.4	\[ \left(-1 \cdot x.im - 2 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) \]
      rational.json-simplify-2 [=>] 0.4	\[ \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(-1 \cdot x.im - 2 \cdot x.im\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ \left(x.re \cdot x.im\right) \cdot \left(\color{blue}{x.im \cdot -1} - 2 \cdot x.im\right) \]
      rational.json-simplify-52 [=>] 0.4	\[ \left(x.re \cdot x.im\right) \cdot \color{blue}{\left(x.im \cdot \left(-1 - 2\right)\right)} \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(\left(-1 - 2\right) \cdot \left(x.re \cdot x.im\right)\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ x.im \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \left(-1 - 2\right)\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ x.im \cdot \left(\color{blue}{\left(x.im \cdot x.re\right)} \cdot \left(-1 - 2\right)\right) \]
      metadata-eval [=>] 0.4	\[ x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot \color{blue}{-3}\right) \]

   if -8.0000000000000004e152 < x.im < 1e133

   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. Simplified 0.2

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 0.2	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   if 1e133 < x.im

   1. Initial program 47.6

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

   2. Simplified 47.6

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

      [Start] 47.6	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 47.6	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational.json-simplify-2 [=>] 47.6	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational.json-simplify-51 [=>] 47.6	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Applied egg-rr 47.6

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

   4. Simplified 0.4

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

      [Start] 47.6	\[ \left(x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + 0\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-4 [=>] 47.6	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-34 [=>] 47.6	\[ x.re \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \color{blue}{\left(x.im + x.re\right) \cdot \left(\left(x.re - x.im\right) \cdot x.re\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{\left(x.re + x.im\right)} \cdot \left(\left(x.re - x.im\right) \cdot x.re\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   5. Taylor expanded in x.re around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \left(x.re + x.im\right) \cdot \left(-1 \cdot \left(x.re \cdot x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-43 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -1\right)\right)} - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]
      rational.json-simplify-9 [=>] 0.4	\[ \left(x.re + x.im\right) \cdot \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) \]

   7. Applied egg-rr 47.8

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

   8. Simplified 0.3

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

      [Start] 47.8	\[ x.re \cdot \left(x.im \cdot \left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right)\right) + 0 \]
      rational.json-simplify-4 [=>] 47.8	\[ \color{blue}{x.re \cdot \left(x.im \cdot \left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right)\right)} \]
      rational.json-simplify-7 [<=] 47.8	\[ x.re \cdot \color{blue}{\frac{x.im \cdot \left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right)}{1}} \]
      rational.json-simplify-7 [=>] 47.8	\[ x.re \cdot \color{blue}{\left(x.im \cdot \left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right)\right)} \]
      rational.json-simplify-5 [<=] 47.8	\[ x.re \cdot \left(x.im \cdot \left(\left(-\left(x.re + x.im\right)\right) - \color{blue}{\left(\left(x.im + x.im\right) - 0\right)}\right)\right) \]
      rational.json-simplify-45 [<=] 47.8	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(0 - \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right)}\right) \]
      rational.json-simplify-12 [<=] 47.8	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(-\left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right)}\right) \]
      rational.json-simplify-8 [=>] 47.8	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(\left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \cdot -1\right)}\right) \]
      rational.json-simplify-43 [<=] 47.8	\[ x.re \cdot \color{blue}{\left(-1 \cdot \left(x.im \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right)\right)} \]
      rational.json-simplify-43 [<=] 47.8	\[ \color{blue}{\left(x.im \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right) \cdot \left(x.re \cdot -1\right)} \]
      rational.json-simplify-8 [<=] 47.8	\[ \left(x.im \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right) \cdot \color{blue}{\left(-x.re\right)} \]
      rational.json-simplify-2 [=>] 47.8	\[ \color{blue}{\left(-x.re\right) \cdot \left(x.im \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)\right)} \]
      rational.json-simplify-43 [<=] 0.3	\[ \color{blue}{\left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \cdot \left(\left(-x.re\right) \cdot x.im\right)} \]
      rational.json-simplify-2 [<=] 0.3	\[ \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \cdot \color{blue}{\left(x.im \cdot \left(-x.re\right)\right)} \]
      rational.json-simplify-2 [=>] 0.3	\[ \color{blue}{\left(x.im \cdot \left(-x.re\right)\right) \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right)} \]
      rational.json-simplify-8 [=>] 0.3	\[ \left(x.im \cdot \color{blue}{\left(x.re \cdot -1\right)}\right) \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \]
      rational.json-simplify-43 [=>] 0.3	\[ \color{blue}{\left(x.re \cdot \left(-1 \cdot x.im\right)\right)} \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \]
      rational.json-simplify-2 [<=] 0.3	\[ \left(x.re \cdot \color{blue}{\left(x.im \cdot -1\right)}\right) \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \]
      rational.json-simplify-9 [=>] 0.3	\[ \left(x.re \cdot \color{blue}{\left(-x.im\right)}\right) \cdot \left(\left(x.im + x.im\right) - \left(-\left(x.re + x.im\right)\right)\right) \]
      rational.json-simplify-12 [=>] 0.3	\[ \left(x.re \cdot \left(-x.im\right)\right) \cdot \left(\left(x.im + x.im\right) - \color{blue}{\left(0 - \left(x.re + x.im\right)\right)}\right) \]

   9. Taylor expanded in x.re around 0 0.3

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

   10. Simplified 0.3

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

       [Start] 0.3	\[ \left(x.re \cdot \left(-x.im\right)\right) \cdot \left(3 \cdot x.im\right) \]
       rational.json-simplify-2 [<=] 0.3	\[ \left(x.re \cdot \left(-x.im\right)\right) \cdot \color{blue}{\left(x.im \cdot 3\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{+152}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 10^{+133}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(-x.im\right)\right) \cdot \left(x.im \cdot 3\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1288

\[\begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{+153}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{elif}\;x.im \leq 8.1 \cdot 10^{+93}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + x.im\right) \cdot \left(x.re \cdot \left(-x.im\right)\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1224

\[\begin{array}{l} t_0 := x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{if}\;x.im \leq -6.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 3.1 \cdot 10^{+85}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

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

Alternative 4
Error	19.6
Cost	896

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

Alternative 5
Error	19.6
Cost	640

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

Alternative 6
Error	19.7
Cost	448

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

Alternative 7
Error	46.4
Cost	384

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

Reproduce

herbie shell --seed 2023066 
(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)))