math.cube on complex, imaginary part

Average Error: 7.5 → 0.4

Time: 53.5s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 1.8 \cdot 10^{+56}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

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 -9.5e+153)
   (* (+ (* x.re (+ x.im x.im)) (* x.im (- x.re x.im))) x.re)
   (if (<= x.re 1.8e+56)
     (+
      (* x.im (- (* x.re (+ x.re x.re)) (* x.im x.im)))
      (* x.re (* x.re x.im)))
     (* (+ x.re (+ x.re (- x.re x.im))) (* x.re 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_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 <= -9.5e+153) {
		tmp = ((x_46_re * (x_46_im + x_46_im)) + (x_46_im * (x_46_re - x_46_im))) * x_46_re;
	} else if (x_46_re <= 1.8e+56) {
		tmp = (x_46_im * ((x_46_re * (x_46_re + x_46_re)) - (x_46_im * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re + (x_46_re + (x_46_re - x_46_im))) * (x_46_re * 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_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 <= (-9.5d+153)) then
        tmp = ((x_46re * (x_46im + x_46im)) + (x_46im * (x_46re - x_46im))) * x_46re
    else if (x_46re <= 1.8d+56) then
        tmp = (x_46im * ((x_46re * (x_46re + x_46re)) - (x_46im * x_46im))) + (x_46re * (x_46re * x_46im))
    else
        tmp = (x_46re + (x_46re + (x_46re - x_46im))) * (x_46re * 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_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 <= -9.5e+153) {
		tmp = ((x_46_re * (x_46_im + x_46_im)) + (x_46_im * (x_46_re - x_46_im))) * x_46_re;
	} else if (x_46_re <= 1.8e+56) {
		tmp = (x_46_im * ((x_46_re * (x_46_re + x_46_re)) - (x_46_im * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else {
		tmp = (x_46_re + (x_46_re + (x_46_re - x_46_im))) * (x_46_re * 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_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 <= -9.5e+153:
		tmp = ((x_46_re * (x_46_im + x_46_im)) + (x_46_im * (x_46_re - x_46_im))) * x_46_re
	elif x_46_re <= 1.8e+56:
		tmp = (x_46_im * ((x_46_re * (x_46_re + x_46_re)) - (x_46_im * x_46_im))) + (x_46_re * (x_46_re * x_46_im))
	else:
		tmp = (x_46_re + (x_46_re + (x_46_re - x_46_im))) * (x_46_re * 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_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 <= -9.5e+153)
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_im + x_46_im)) + Float64(x_46_im * Float64(x_46_re - x_46_im))) * x_46_re);
	elseif (x_46_re <= 1.8e+56)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * Float64(x_46_re + x_46_re)) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_re + Float64(x_46_re - x_46_im))) * Float64(x_46_re * 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_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 <= -9.5e+153)
		tmp = ((x_46_re * (x_46_im + x_46_im)) + (x_46_im * (x_46_re - x_46_im))) * x_46_re;
	elseif (x_46_re <= 1.8e+56)
		tmp = (x_46_im * ((x_46_re * (x_46_re + x_46_re)) - (x_46_im * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	else
		tmp = (x_46_re + (x_46_re + (x_46_re - x_46_im))) * (x_46_re * 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$46$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, -9.5e+153], N[(N[(N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[x$46$re, 1.8e+56], N[(N[(x$46$im * N[(N[(x$46$re * N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(x$46$re + N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]]

Target

Original	7.5
Target	0.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if x.re < -9.4999999999999995e153

   1. Initial program 63.9

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

   2. Simplified 0.4

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

      [Start] 63.9	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 63.9	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-111 [=>] 63.9	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-50 [=>] 0.4	\[ \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-3 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + x.im\right)} \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + x.re \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational_best-simplify-63 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Taylor expanded in x.re around inf 0.4

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

   4. Applied egg-rr 0.4

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

   if -9.4999999999999995e153 < x.re < 1.79999999999999999e56

   1. Initial program 0.2

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

   2. Simplified 0.2

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

      [Start] 0.2	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 0.2	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-7 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \color{blue}{\left(x.re \cdot 1\right)} \]
      metadata-eval [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(x.re \cdot \color{blue}{\frac{-2}{-2}}\right) \]
      metadata-eval [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(x.re \cdot \frac{\color{blue}{-1 + -1}}{-2}\right) \]
      metadata-eval [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \left(x.re \cdot \frac{-1 + -1}{\color{blue}{-1 + -1}}\right) \]
      rational_best-simplify-1 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \color{blue}{\left(\frac{-1 + -1}{-1 + -1} \cdot x.re\right)} \]
      rational_best-simplify-50 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(\frac{-1 + -1}{-1 + -1} \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)\right)} \]
      rational_best-simplify-1 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot \frac{-1 + -1}{-1 + -1}\right)} \]
      rational_best-simplify-55 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \color{blue}{\left(\left(-1 + -1\right) \cdot \frac{x.re \cdot x.im + x.im \cdot x.re}{-1 + -1}\right)} \]
      rational_best-simplify-1 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(-1 + -1\right) \cdot \frac{x.re \cdot x.im + \color{blue}{x.re \cdot x.im}}{-1 + -1}\right) \]
      rational_best-simplify-108 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(-1 + -1\right) \cdot \color{blue}{\frac{x.re \cdot x.im}{-1}}\right) \]
      rational_best-simplify-55 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \color{blue}{\left(\left(x.re \cdot x.im\right) \cdot \frac{-1 + -1}{-1}\right)} \]
      metadata-eval [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot \frac{\color{blue}{-2}}{-1}\right) \]
      metadata-eval [=>] 0.2	\[ x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot \color{blue}{2}\right) \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - \left(-x.im \cdot \left(x.re \cdot x.re\right)\right) \]
      rational_best-simplify-11 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - \color{blue}{\left(x.im \cdot \left(x.re \cdot x.re\right)\right) \cdot -1} \]
      rational_best-simplify-1 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - \color{blue}{-1 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)} \]
      rational_best-simplify-50 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - -1 \cdot \color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]
      rational_best-simplify-1 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - -1 \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \]
      rational_best-simplify-1 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - -1 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} \]
      rational_best-simplify-50 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - \color{blue}{x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot -1\right)} \]
      rational_best-simplify-1 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(-1 \cdot \left(x.im \cdot x.re\right)\right)} \]
      rational_best-simplify-1 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - x.re \cdot \left(-1 \cdot \color{blue}{\left(x.re \cdot x.im\right)}\right) \]
      rational_best-simplify-50 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - x.re \cdot \color{blue}{\left(x.im \cdot \left(x.re \cdot -1\right)\right)} \]
      rational_best-simplify-11 [<=] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) - x.re \cdot \left(x.im \cdot \color{blue}{\left(-x.re\right)}\right) \]

   5. Applied egg-rr 0.2

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

   6. Simplified 0.2

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

      [Start] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re\right) \]
      rational_best-simplify-50 [=>] 0.2	\[ x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im\right)} \]

   if 1.79999999999999999e56 < x.re

   1. Initial program 26.1

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

   2. Simplified 0.4

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

      [Start] 26.1	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 26.1	\[ \color{blue}{x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-111 [=>] 26.1	\[ x.im \cdot \color{blue}{\left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-50 [=>] 0.4	\[ \color{blue}{\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-3 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\color{blue}{\left(x.re + x.im\right)} \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + \color{blue}{x.re \cdot \left(x.re \cdot x.im + x.im \cdot x.re\right)} \]
      rational_best-simplify-1 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + x.re \cdot \left(\color{blue}{x.im \cdot x.re} + x.im \cdot x.re\right) \]
      rational_best-simplify-63 [=>] 0.4	\[ \left(x.re - x.im\right) \cdot \left(\left(x.re + x.im\right) \cdot x.im\right) + x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \]

   3. Taylor expanded in x.re around inf 1.8

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

   4. Applied egg-rr 1.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 1.8 \cdot 10^{+56}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(x.re + x.re\right) - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x.re \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 10^{+131}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5.4 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{+72}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 4 - \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	1088

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

Alternative 4
Error	0.4
Cost	968

\[\begin{array}{l} t_0 := x.re \cdot \left(x.im \cdot \left(\left(x.re - x.im\right) + \left(x.re + x.re\right)\right)\right)\\ \mathbf{if}\;x.re \leq -6 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.6
Cost	968

\[\begin{array}{l} t_0 := \left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \mathbf{if}\;x.re \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 3.6 \cdot 10^{+72}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.re \leq -5.2 \cdot 10^{+153}:\\ \;\;\;\;\left(x.re \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.re - x.im\right)\right) \cdot x.re\\ \mathbf{elif}\;x.re \leq 5.3 \cdot 10^{+72}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \left(x.re + \left(x.re - x.im\right)\right)\right) \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 7
Error	7.6
Cost	704

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

Alternative 8
Error	7.6
Cost	704

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

Reproduce

herbie shell --seed 2023099 
(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)))