math.cube on complex, real part

Average Error: 7.3 → 0.3

Time: 8.6s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \left(-x.im\right)\\ \end{array} \]

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 -1.4e+154)
   (* x.im (* x.re (* -3.0 x.im)))
   (if (<= x.im 1.65e+70)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* (+ x.im x.im) (* x.re x.im)))
     (* (* x.im (* x.re 3.0)) (- 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 <= -1.4e+154) {
		tmp = x_46_im * (x_46_re * (-3.0 * x_46_im));
	} else if (x_46_im <= 1.65e+70) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im + x_46_im) * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_im * (x_46_re * 3.0)) * -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 <= (-1.4d+154)) then
        tmp = x_46im * (x_46re * ((-3.0d0) * x_46im))
    else if (x_46im <= 1.65d+70) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - ((x_46im + x_46im) * (x_46re * x_46im))
    else
        tmp = (x_46im * (x_46re * 3.0d0)) * -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 <= -1.4e+154) {
		tmp = x_46_im * (x_46_re * (-3.0 * x_46_im));
	} else if (x_46_im <= 1.65e+70) {
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im + x_46_im) * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_im * (x_46_re * 3.0)) * -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 <= -1.4e+154:
		tmp = x_46_im * (x_46_re * (-3.0 * x_46_im))
	elif x_46_im <= 1.65e+70:
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im + x_46_im) * (x_46_re * x_46_im))
	else:
		tmp = (x_46_im * (x_46_re * 3.0)) * -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 <= -1.4e+154)
		tmp = Float64(x_46_im * Float64(x_46_re * Float64(-3.0 * x_46_im)));
	elseif (x_46_im <= 1.65e+70)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) - Float64(Float64(x_46_im + x_46_im) * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_im * Float64(x_46_re * 3.0)) * Float64(-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 <= -1.4e+154)
		tmp = x_46_im * (x_46_re * (-3.0 * x_46_im));
	elseif (x_46_im <= 1.65e+70)
		tmp = (x_46_re * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) - ((x_46_im + x_46_im) * (x_46_re * x_46_im));
	else
		tmp = (x_46_im * (x_46_re * 3.0)) * -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, -1.4e+154], N[(x$46$im * N[(x$46$re * N[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.65e+70], 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[(N[(x$46$im + x$46$im), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision] * (-x$46$im)), $MachinePrecision]]]

Target

Original	7.3
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x.im < -1.4e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 64.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-3 [<=] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-62 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot x.im - \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-111 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-113 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(\color{blue}{x.re \cdot \left(x.im \cdot x.im\right)} - x.im \cdot \left(-x.re \cdot x.im\right)\right) \]
      rational_best-simplify-3 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \left(-\color{blue}{x.im \cdot x.re}\right)\right) \]
      rational_best-simplify-52 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-113 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-52 [<=] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(-x.im \cdot x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot x.im - \left(-x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-52 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot x.im - \color{blue}{x.im \cdot \left(-x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im - \left(-x.im\right)\right)\right)} \]
      rational_best-simplify-62 [<=] 64.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot \color{blue}{\left(x.im + x.im\right)}\right) \]

   3. Applied egg-rr 64.0

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

   4. Applied egg-rr 64.0

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

   5. Simplified 64.0

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

      [Start] 64.0	\[ \left(x.re \cdot x.re + \left(x.im \cdot \left(\left(-x.im\right) - x.im\right) - x.im \cdot x.im\right)\right) \cdot x.re \]
      rational_best-simplify-110 [=>] 64.0	\[ \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(\left(\left(-x.im\right) - x.im\right) - x.im\right)}\right) \cdot x.re \]

   6. Taylor expanded in x.re around 0 64.0

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

   7. Simplified 64.0

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

      [Start] 64.0	\[ \left(-1 \cdot \left(\left(2 \cdot x.im + x.im\right) \cdot x.im\right)\right) \cdot x.re \]
      rational_best-simplify-113 [=>] 64.0	\[ \color{blue}{\left(\left(2 \cdot x.im + x.im\right) \cdot \left(-1 \cdot x.im\right)\right)} \cdot x.re \]
      rational_best-simplify-3 [=>] 64.0	\[ \left(\left(2 \cdot x.im + x.im\right) \cdot \color{blue}{\left(x.im \cdot -1\right)}\right) \cdot x.re \]
      rational_best-simplify-18 [<=] 64.0	\[ \left(\left(2 \cdot x.im + x.im\right) \cdot \color{blue}{\left(-x.im\right)}\right) \cdot x.re \]
      rational_best-simplify-62 [=>] 64.0	\[ \left(\color{blue}{\left(x.im - \left(-2 \cdot x.im\right)\right)} \cdot \left(-x.im\right)\right) \cdot x.re \]
      rational_best-simplify-52 [=>] 64.0	\[ \left(\left(x.im - \color{blue}{x.im \cdot \left(-2\right)}\right) \cdot \left(-x.im\right)\right) \cdot x.re \]
      metadata-eval [=>] 64.0	\[ \left(\left(x.im - x.im \cdot \color{blue}{-2}\right) \cdot \left(-x.im\right)\right) \cdot x.re \]
      rational_best-simplify-56 [<=] 64.0	\[ \color{blue}{\left(x.im \cdot \left(-\left(x.im - x.im \cdot -2\right)\right)\right)} \cdot x.re \]
      metadata-eval [<=] 64.0	\[ \left(x.im \cdot \left(-\left(x.im - x.im \cdot \color{blue}{\left(-2\right)}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-52 [<=] 64.0	\[ \left(x.im \cdot \left(-\left(x.im - \color{blue}{\left(-2 \cdot x.im\right)}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-62 [<=] 64.0	\[ \left(x.im \cdot \left(-\color{blue}{\left(2 \cdot x.im + x.im\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 64.0	\[ \left(x.im \cdot \left(-\color{blue}{\left(x.im + 2 \cdot x.im\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-62 [=>] 64.0	\[ \left(x.im \cdot \left(-\color{blue}{\left(2 \cdot x.im - \left(-x.im\right)\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-3 [=>] 64.0	\[ \left(x.im \cdot \left(-\left(\color{blue}{x.im \cdot 2} - \left(-x.im\right)\right)\right)\right) \cdot x.re \]
      rational_best-simplify-18 [=>] 64.0	\[ \left(x.im \cdot \left(-\left(x.im \cdot 2 - \color{blue}{x.im \cdot -1}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-110 [=>] 64.0	\[ \left(x.im \cdot \left(-\color{blue}{x.im \cdot \left(2 - -1\right)}\right)\right) \cdot x.re \]
      metadata-eval [=>] 64.0	\[ \left(x.im \cdot \left(-x.im \cdot \color{blue}{3}\right)\right) \cdot x.re \]

   8. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;0 \ne 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot -3\right) \cdot x.re\right)\\ } \end{array}} \]

   9. Simplified 0.4

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

      [Start] 0.4	\[ \begin{array}{l} \mathbf{if}\;0 \ne 0:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot -3\right) \cdot x.re\right)\\ \end{array} \]
      rational_best-simplify-8 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(\left(x.im \cdot -3\right) \cdot x.re\right)} \]
      rational_best-simplify-3 [=>] 0.4	\[ x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]
      rational_best-simplify-3 [=>] 0.4	\[ x.im \cdot \left(x.re \cdot \color{blue}{\left(-3 \cdot x.im\right)}\right) \]

   if -1.4e154 < x.im < 1.65000000000000008e70

   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

      \[\leadsto \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      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_best-simplify-3 [=>] 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_best-simplify-3 [=>] 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_best-simplify-3 [<=] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-62 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot x.im - \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-111 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-113 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(\color{blue}{x.re \cdot \left(x.im \cdot x.im\right)} - x.im \cdot \left(-x.re \cdot x.im\right)\right) \]
      rational_best-simplify-3 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \left(-\color{blue}{x.im \cdot x.re}\right)\right) \]
      rational_best-simplify-52 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-113 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-52 [<=] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(-x.im \cdot x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot x.im - \left(-x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-52 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot x.im - \color{blue}{x.im \cdot \left(-x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im - \left(-x.im\right)\right)\right)} \]
      rational_best-simplify-62 [<=] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot \color{blue}{\left(x.im + x.im\right)}\right) \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) + 0\right) \]
      rational_best-simplify-62 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(0 - \left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\right)} \]
      rational_best-simplify-19 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(-\left(-x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\right)\right)} \]
      rational_best-simplify-51 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-113 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best-simplify-3 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      rational_best-simplify-113 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im + x.im\right) \cdot \left(x.re \cdot x.im\right)} \]

   if 1.65000000000000008e70 < x.im

   1. Initial program 27.0

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

   2. Simplified 27.1

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

      [Start] 27.0	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 27.0	\[ \color{blue}{x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best-simplify-3 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-3 [<=] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \]
      rational_best-simplify-62 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot x.im - \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-111 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.re \cdot x.im\right) - x.im \cdot \left(-x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-113 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(\color{blue}{x.re \cdot \left(x.im \cdot x.im\right)} - x.im \cdot \left(-x.re \cdot x.im\right)\right) \]
      rational_best-simplify-3 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \left(-\color{blue}{x.im \cdot x.re}\right)\right) \]
      rational_best-simplify-52 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.re \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-113 [=>] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)}\right) \]
      rational_best-simplify-52 [<=] 27.0	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.re \cdot \left(x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(-x.im \cdot x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 27.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(x.im \cdot x.im - \left(-x.im \cdot x.im\right)\right)} \]
      rational_best-simplify-52 [=>] 27.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot x.im - \color{blue}{x.im \cdot \left(-x.im\right)}\right) \]
      rational_best-simplify-110 [=>] 27.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(x.im \cdot \left(x.im - \left(-x.im\right)\right)\right)} \]
      rational_best-simplify-62 [<=] 27.1	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot \color{blue}{\left(x.im + x.im\right)}\right) \]

   3. Applied egg-rr 27.2

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

   4. Applied egg-rr 27.2

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

   5. Simplified 27.2

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

      [Start] 27.2	\[ \left(x.re \cdot x.re + \left(x.im \cdot \left(\left(-x.im\right) - x.im\right) - x.im \cdot x.im\right)\right) \cdot x.re \]
      rational_best-simplify-110 [=>] 27.2	\[ \left(x.re \cdot x.re + \color{blue}{x.im \cdot \left(\left(\left(-x.im\right) - x.im\right) - x.im\right)}\right) \cdot x.re \]

   6. Taylor expanded in x.re around 0 27.7

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

   7. Simplified 27.7

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

      [Start] 27.7	\[ \left(-1 \cdot \left(\left(2 \cdot x.im + x.im\right) \cdot x.im\right)\right) \cdot x.re \]
      rational_best-simplify-113 [=>] 27.7	\[ \color{blue}{\left(\left(2 \cdot x.im + x.im\right) \cdot \left(-1 \cdot x.im\right)\right)} \cdot x.re \]
      rational_best-simplify-3 [=>] 27.7	\[ \left(\left(2 \cdot x.im + x.im\right) \cdot \color{blue}{\left(x.im \cdot -1\right)}\right) \cdot x.re \]
      rational_best-simplify-18 [<=] 27.7	\[ \left(\left(2 \cdot x.im + x.im\right) \cdot \color{blue}{\left(-x.im\right)}\right) \cdot x.re \]
      rational_best-simplify-62 [=>] 27.7	\[ \left(\color{blue}{\left(x.im - \left(-2 \cdot x.im\right)\right)} \cdot \left(-x.im\right)\right) \cdot x.re \]
      rational_best-simplify-52 [=>] 27.7	\[ \left(\left(x.im - \color{blue}{x.im \cdot \left(-2\right)}\right) \cdot \left(-x.im\right)\right) \cdot x.re \]
      metadata-eval [=>] 27.7	\[ \left(\left(x.im - x.im \cdot \color{blue}{-2}\right) \cdot \left(-x.im\right)\right) \cdot x.re \]
      rational_best-simplify-56 [<=] 27.7	\[ \color{blue}{\left(x.im \cdot \left(-\left(x.im - x.im \cdot -2\right)\right)\right)} \cdot x.re \]
      metadata-eval [<=] 27.7	\[ \left(x.im \cdot \left(-\left(x.im - x.im \cdot \color{blue}{\left(-2\right)}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-52 [<=] 27.7	\[ \left(x.im \cdot \left(-\left(x.im - \color{blue}{\left(-2 \cdot x.im\right)}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-62 [<=] 27.7	\[ \left(x.im \cdot \left(-\color{blue}{\left(2 \cdot x.im + x.im\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 27.7	\[ \left(x.im \cdot \left(-\color{blue}{\left(x.im + 2 \cdot x.im\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-62 [=>] 27.7	\[ \left(x.im \cdot \left(-\color{blue}{\left(2 \cdot x.im - \left(-x.im\right)\right)}\right)\right) \cdot x.re \]
      rational_best-simplify-3 [=>] 27.7	\[ \left(x.im \cdot \left(-\left(\color{blue}{x.im \cdot 2} - \left(-x.im\right)\right)\right)\right) \cdot x.re \]
      rational_best-simplify-18 [=>] 27.7	\[ \left(x.im \cdot \left(-\left(x.im \cdot 2 - \color{blue}{x.im \cdot -1}\right)\right)\right) \cdot x.re \]
      rational_best-simplify-110 [=>] 27.7	\[ \left(x.im \cdot \left(-\color{blue}{x.im \cdot \left(2 - -1\right)}\right)\right) \cdot x.re \]
      metadata-eval [=>] 27.7	\[ \left(x.im \cdot \left(-x.im \cdot \color{blue}{3}\right)\right) \cdot x.re \]

   8. Applied egg-rr 0.9

      \[\leadsto \color{blue}{0 - x.im \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]

   9. Simplified 0.9

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

      [Start] 0.9	\[ 0 - x.im \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right) \]
      rational_best-simplify-19 [=>] 0.9	\[ \color{blue}{-x.im \cdot \left(x.im \cdot \left(3 \cdot x.re\right)\right)} \]
      rational_best-simplify-52 [=>] 0.9	\[ \color{blue}{\left(x.im \cdot \left(3 \cdot x.re\right)\right) \cdot \left(-x.im\right)} \]
      rational_best-simplify-3 [=>] 0.9	\[ \left(x.im \cdot \color{blue}{\left(x.re \cdot 3\right)}\right) \cdot \left(-x.im\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{+70}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+104}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.55 \cdot 10^{+70}:\\ \;\;\;\;\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \left(-x.im\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1216

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

Alternative 3
Error	0.5
Cost	968

\[\begin{array}{l} t_0 := \left(x.im \cdot \left(x.re \cdot 3\right)\right) \cdot \left(-x.im\right)\\ \mathbf{if}\;x.im \leq -5.6 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{+69}:\\ \;\;\;\;\left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right) \cdot x.re\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.4
Cost	512

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

Alternative 5
Error	19.4
Cost	448

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

Reproduce

herbie shell --seed 2023102 
(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)))