math.cube on complex, real part

Average Error: 7.8 → 0.3

Time: 10.5s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \end{array} \]

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 -2.5e+61)
   (* x.im (* x.im (* x.re -3.0)))
   (if (<= x.im 2.4e+141)
     (-
      (* x.re (- (* x.re x.re) (* x.im x.im)))
      (* x.im (* x.re (+ x.im 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 <= -2.5e+61) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 2.4e+141) {
		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_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 <= (-2.5d+61)) then
        tmp = x_46im * (x_46im * (x_46re * (-3.0d0)))
    else if (x_46im <= 2.4d+141) then
        tmp = (x_46re * ((x_46re * x_46re) - (x_46im * x_46im))) - (x_46im * (x_46re * (x_46im + 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 <= -2.5e+61) {
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	} else if (x_46_im <= 2.4e+141) {
		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_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 <= -2.5e+61:
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0))
	elif x_46_im <= 2.4e+141:
		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_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 <= -2.5e+61)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(x_46_re * -3.0)));
	elseif (x_46_im <= 2.4e+141)
		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(x_46_im * Float64(x_46_re * Float64(-3.0 * 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 <= -2.5e+61)
		tmp = x_46_im * (x_46_im * (x_46_re * -3.0));
	elseif (x_46_im <= 2.4e+141)
		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_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, -2.5e+61], N[(x$46$im * N[(x$46$im * N[(x$46$re * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.4e+141], 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[(x$46$im * N[(x$46$re * N[(-3.0 * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.8
Target	0.2
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 < -2.50000000000000009e61

   1. Initial program 26.1

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

   2. Simplified 26.2

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

      [Start] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 26.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 26.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.re \cdot \color{blue}{\left(x.im \cdot x.im + x.im \cdot x.im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 26.2	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im + x.im \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 26.2	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.im + x.im\right)}\right) \]

   3. Applied egg-rr 26.2

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

   4. Simplified 26.2

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

      [Start] 26.2	\[ x.re \cdot \left(\left(x.im \cdot x.im + \left(x.im \cdot \left(x.im + x.im\right) - x.re \cdot x.re\right)\right) \cdot -1\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 26.2	\[ x.re \cdot \color{blue}{\left(-\left(x.im \cdot x.im + \left(x.im \cdot \left(x.im + x.im\right) - x.re \cdot x.re\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 26.2	\[ x.re \cdot \left(-\color{blue}{\left(\left(x.im \cdot \left(x.im + x.im\right) + x.im \cdot x.im\right) - x.re \cdot x.re\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 26.2	\[ x.re \cdot \left(-\left(\left(\color{blue}{\left(x.im + x.im\right) \cdot x.im} + x.im \cdot x.im\right) - x.re \cdot x.re\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 26.2	\[ x.re \cdot \left(-\left(\color{blue}{x.im \cdot \left(\left(x.im + x.im\right) + x.im\right)} - x.re \cdot x.re\right)\right) \]

   5. Taylor expanded in x.re around 0 27.0

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

   6. Simplified 27.0

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

      [Start] 27.0	\[ -1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.0	\[ \color{blue}{x.re \cdot \left(-1 \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.0	\[ x.re \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot \left(x.im + 2 \cdot x.im\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.0	\[ x.re \cdot \color{blue}{\left(x.im \cdot \left(-1 \cdot \left(x.im + 2 \cdot x.im\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 27.0	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \color{blue}{\left(2 \cdot x.im + x.im\right)}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [<=] 27.0	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \left(2 \cdot x.im + \color{blue}{x.im \cdot 1}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 27.0	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot \left(2 + 1\right)\right)}\right)\right) \]
      metadata-eval [=>] 27.0	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \left(x.im \cdot \color{blue}{3}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 27.0	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(\left(x.im \cdot 3\right) \cdot -1\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 27.0	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(-x.im \cdot 3\right)}\right) \]

   7. Applied egg-rr 1.2

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

   8. Simplified 26.9

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

      [Start] 1.2	\[ x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right) + 0 \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 1.2	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 1.2	\[ x.im \cdot \color{blue}{\left(-3 \cdot \left(x.re \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.2	\[ \color{blue}{-3 \cdot \left(x.im \cdot \left(x.re \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [<=] 26.9	\[ -3 \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right)} \]

   9. Applied egg-rr 27.0

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

   10. Simplified 1.2

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

       [Start] 27.0	\[ x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right) + 0 \]
       rational_best_oopsla_all_46_json_45_simplify-85 [=>] 27.0	\[ \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im \cdot -3\right)\right)} \]
       rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.2	\[ \color{blue}{x.im \cdot \left(x.re \cdot \left(x.im \cdot -3\right)\right)} \]
       rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.2	\[ x.im \cdot \color{blue}{\left(x.im \cdot \left(x.re \cdot -3\right)\right)} \]

   if -2.50000000000000009e61 < x.im < 2.39999999999999997e141

   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_best_oopsla_all_46_json_45_simplify-74 [=>] 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_oopsla_all_46_json_45_simplify-74 [=>] 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_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 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 2.39999999999999997e141 < x.im

   1. Initial program 55.7

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

   2. Simplified 55.8

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

      [Start] 55.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.im \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 55.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.im \cdot x.re + x.re \cdot x.im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 55.7	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.im \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.8	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(x.im \cdot \left(x.im + x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 55.8	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - x.re \cdot \color{blue}{\left(x.im \cdot x.im + x.im \cdot x.im\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-102 [=>] 55.8	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im \cdot x.im + x.im \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 55.8	\[ x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \color{blue}{x.im \cdot \left(x.im + x.im\right)}\right) \]

   3. Applied egg-rr 55.8

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

   4. Simplified 55.8

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

      [Start] 55.8	\[ x.re \cdot \left(\left(x.im \cdot x.im + \left(x.im \cdot \left(x.im + x.im\right) - x.re \cdot x.re\right)\right) \cdot -1\right) \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 55.8	\[ x.re \cdot \color{blue}{\left(-\left(x.im \cdot x.im + \left(x.im \cdot \left(x.im + x.im\right) - x.re \cdot x.re\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 55.8	\[ x.re \cdot \left(-\color{blue}{\left(\left(x.im \cdot \left(x.im + x.im\right) + x.im \cdot x.im\right) - x.re \cdot x.re\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.8	\[ x.re \cdot \left(-\left(\left(\color{blue}{\left(x.im + x.im\right) \cdot x.im} + x.im \cdot x.im\right) - x.re \cdot x.re\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 55.8	\[ x.re \cdot \left(-\left(\color{blue}{x.im \cdot \left(\left(x.im + x.im\right) + x.im\right)} - x.re \cdot x.re\right)\right) \]

   5. Taylor expanded in x.re around 0 55.8

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

   6. Simplified 55.8

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

      [Start] 55.8	\[ -1 \cdot \left(x.re \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.8	\[ \color{blue}{x.re \cdot \left(-1 \cdot \left(\left(x.im + 2 \cdot x.im\right) \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 55.8	\[ x.re \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot \left(x.im + 2 \cdot x.im\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 55.8	\[ x.re \cdot \color{blue}{\left(x.im \cdot \left(-1 \cdot \left(x.im + 2 \cdot x.im\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 55.8	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \color{blue}{\left(2 \cdot x.im + x.im\right)}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [<=] 55.8	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \left(2 \cdot x.im + \color{blue}{x.im \cdot 1}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 55.8	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \color{blue}{\left(x.im \cdot \left(2 + 1\right)\right)}\right)\right) \]
      metadata-eval [=>] 55.8	\[ x.re \cdot \left(x.im \cdot \left(-1 \cdot \left(x.im \cdot \color{blue}{3}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 55.8	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(\left(x.im \cdot 3\right) \cdot -1\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-94 [<=] 55.8	\[ x.re \cdot \left(x.im \cdot \color{blue}{\left(-x.im \cdot 3\right)}\right) \]

   7. Applied egg-rr 0.4

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

   8. Simplified 0.3

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

      [Start] 0.4	\[ x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right) + 0 \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.4	\[ \color{blue}{x.im \cdot \left(\left(x.re \cdot x.im\right) \cdot -3\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ x.im \cdot \color{blue}{\left(-3 \cdot \left(x.re \cdot x.im\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.3	\[ x.im \cdot \color{blue}{\left(x.re \cdot \left(-3 \cdot x.im\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{+61}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) - x.im \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-3 \cdot x.im\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1152

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

Alternative 2
Error	0.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -6.5 \cdot 10^{+79}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(x.re \cdot -3\right)\right)\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{+26}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re - x.im \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-3 \cdot \left(-\left(x.re \cdot \left(-x.im\right)\right) \cdot x.im\right)\\ \end{array} \]

Alternative 3
Error	19.0
Cost	576

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

Alternative 4
Error	26.6
Cost	448

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

Alternative 5
Error	19.0
Cost	448

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

Alternative 6
Error	19.0
Cost	448

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

Alternative 7
Error	19.0
Cost	448

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

Reproduce

herbie shell --seed 2023090 
(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)))