math.cube on complex, real part

Percentage Accurate: 82.5% → 95.9%

Time: 5.1s

Precision: binary64

Cost: 968

• 44.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

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 -2.05e+133)
   (* (* x.im (* x.im x.re)) -3.0)
   (if (<= x.im 7.6e+153)
     (* x.re (+ (* x.re x.re) (* x.im (* x.im -3.0))))
     (* x.im (* (* x.im x.re) -3.0)))))

C

double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_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 <= -2.05e+133) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = (((x_46re * x_46re) - (x_46im * x_46im)) * x_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 <= (-2.05d+133)) then
        tmp = (x_46im * (x_46im * x_46re)) * (-3.0d0)
    else if (x_46im <= 7.6d+153) then
        tmp = x_46re * ((x_46re * x_46re) + (x_46im * (x_46im * (-3.0d0))))
    else
        tmp = x_46im * ((x_46im * x_46re) * (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_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 <= -2.05e+133) {
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	} else if (x_46_im <= 7.6e+153) {
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	} else {
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	return (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_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 <= -2.05e+133:
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0
	elif x_46_im <= 7.6e+153:
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)))
	else:
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) * x_46_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 <= -2.05e+133)
		tmp = Float64(Float64(x_46_im * Float64(x_46_im * x_46_re)) * -3.0);
	elseif (x_46_im <= 7.6e+153)
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * Float64(x_46_im * -3.0))));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_im * x_46_re) * -3.0));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (((x_46_re * x_46_re) - (x_46_im * x_46_im)) * x_46_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 <= -2.05e+133)
		tmp = (x_46_im * (x_46_im * x_46_re)) * -3.0;
	elseif (x_46_im <= 7.6e+153)
		tmp = x_46_re * ((x_46_re * x_46_re) + (x_46_im * (x_46_im * -3.0)));
	else
		tmp = x_46_im * ((x_46_im * x_46_re) * -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * x$46$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, -2.05e+133], N[(N[(x$46$im * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[x$46$im, 7.6e+153], N[(x$46$re * N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * N[(x$46$im * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$im * x$46$re), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	82.5%
Target	86.8%
Herbie	95.9%

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

Derivation

1. Split input into 3 regimes

2. if x.im < -2.05000000000000002e133

   1. Initial program 64.3%

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

   2. Simplified 73.5%

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

      [Start] 64.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [<=] 64.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]
      distribute-lft-out [=>] 64.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
      associate-*l* [=>] 64.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 64.3%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      distribute-rgt-out-- [=>] 64.2%	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      associate--l- [=>] 64.2%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      associate--l- [<=] 64.2%	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      sub-neg [=>] 64.2%	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      associate--l+ [=>] 64.2%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      fma-udef [<=] 73.5%	\[ x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, \left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)}\right) \]
      associate-*r* [<=] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 - 2\right)\right)}\right) \]
      metadata-eval [=>] 73.5%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \]

   3. Taylor expanded in x.re around 0 73.6%

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

   4. Simplified 73.5%

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

      [Start] 73.6%	\[ -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
      associate-*r* [=>] 73.5%	\[ \color{blue}{\left(-3 \cdot x.re\right) \cdot {x.im}^{2}} \]
      *-commutative [=>] 73.5%	\[ \color{blue}{{x.im}^{2} \cdot \left(-3 \cdot x.re\right)} \]
      *-commutative [=>] 73.5%	\[ {x.im}^{2} \cdot \color{blue}{\left(x.re \cdot -3\right)} \]
      metadata-eval [<=] 73.5%	\[ {x.im}^{2} \cdot \left(x.re \cdot \color{blue}{\left(-2 + -1\right)}\right) \]
      distribute-rgt-out [<=] 73.5%	\[ {x.im}^{2} \cdot \color{blue}{\left(-2 \cdot x.re + -1 \cdot x.re\right)} \]
      distribute-lft-in [=>] 73.6%	\[ \color{blue}{{x.im}^{2} \cdot \left(-2 \cdot x.re\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right)} \]
      metadata-eval [<=] 73.6%	\[ {x.im}^{2} \cdot \left(\color{blue}{\left(-2\right)} \cdot x.re\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-neg-in [<=] 73.6%	\[ {x.im}^{2} \cdot \color{blue}{\left(-2 \cdot x.re\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      count-2 [<=] 73.6%	\[ {x.im}^{2} \cdot \left(-\color{blue}{\left(x.re + x.re\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-neg-in [<=] 73.6%	\[ \color{blue}{\left(-{x.im}^{2} \cdot \left(x.re + x.re\right)\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-out [<=] 73.6%	\[ \left(-\color{blue}{\left(x.re \cdot {x.im}^{2} + x.re \cdot {x.im}^{2}\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-out [=>] 73.6%	\[ \left(-\color{blue}{x.re \cdot \left({x.im}^{2} + {x.im}^{2}\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-neg-in [=>] 73.6%	\[ \color{blue}{x.re \cdot \left(-\left({x.im}^{2} + {x.im}^{2}\right)\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      count-2 [=>] 73.6%	\[ x.re \cdot \left(-\color{blue}{2 \cdot {x.im}^{2}}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-neg-in [=>] 73.6%	\[ x.re \cdot \color{blue}{\left(\left(-2\right) \cdot {x.im}^{2}\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      metadata-eval [=>] 73.6%	\[ x.re \cdot \left(\color{blue}{-2} \cdot {x.im}^{2}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      mul-1-neg [=>] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + {x.im}^{2} \cdot \color{blue}{\left(-x.re\right)} \]
      distribute-rgt-neg-in [<=] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(-{x.im}^{2} \cdot x.re\right)} \]
      distribute-lft-neg-in [=>] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(-{x.im}^{2}\right) \cdot x.re} \]
      unpow2 [=>] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \left(-\color{blue}{x.im \cdot x.im}\right) \cdot x.re \]
      distribute-rgt-neg-out [<=] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \cdot x.re \]
      *-commutative [<=] 73.6%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)} \]

   5. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{{\left(\sqrt{x.re \cdot -3} \cdot x.im\right)}^{2}} \]
      Step-by-step derivation

      [Start] 73.5%	\[ x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right) \]
      add-sqr-sqrt [=>] 35.8%	\[ \color{blue}{\sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)} \cdot \sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)}} \]
      pow2 [=>] 35.8%	\[ \color{blue}{{\left(\sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)}\right)}^{2}} \]
      associate-*r* [=>] 35.8%	\[ {\left(\sqrt{\color{blue}{\left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)}}\right)}^{2} \]
      sqrt-prod [=>] 35.8%	\[ {\color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.im \cdot x.im}\right)}}^{2} \]
      sqrt-prod [=>] 0.0%	\[ {\left(\sqrt{x.re \cdot -3} \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)}\right)}^{2} \]
      add-sqr-sqrt [<=] 46.4%	\[ {\left(\sqrt{x.re \cdot -3} \cdot \color{blue}{x.im}\right)}^{2} \]

   6. Applied egg-rr 93.0%

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

      [Start] 46.4%	\[ {\left(\sqrt{x.re \cdot -3} \cdot x.im\right)}^{2} \]
      unpow2 [=>] 46.4%	\[ \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot x.im\right) \cdot \left(\sqrt{x.re \cdot -3} \cdot x.im\right)} \]
      swap-sqr [=>] 35.7%	\[ \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.re \cdot -3}\right) \cdot \left(x.im \cdot x.im\right)} \]
      add-sqr-sqrt [<=] 73.5%	\[ \color{blue}{\left(x.re \cdot -3\right)} \cdot \left(x.im \cdot x.im\right) \]
      *-commutative [=>] 73.5%	\[ \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(x.re \cdot -3\right)} \]
      associate-*l* [<=] 73.6%	\[ \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot x.re\right) \cdot -3} \]
      *-commutative [<=] 73.6%	\[ \color{blue}{\left(x.re \cdot \left(x.im \cdot x.im\right)\right)} \cdot -3 \]
      *-commutative [=>] 73.6%	\[ \color{blue}{\left(\left(x.im \cdot x.im\right) \cdot x.re\right)} \cdot -3 \]
      associate-*l* [=>] 93.0%	\[ \color{blue}{\left(x.im \cdot \left(x.im \cdot x.re\right)\right)} \cdot -3 \]

   if -2.05000000000000002e133 < x.im < 7.59999999999999933e153

   1. Initial program 91.9%

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

   2. Simplified 99.8%

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

      [Start] 91.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [<=] 91.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]
      distribute-lft-out [=>] 91.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
      associate-*l* [=>] 91.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 91.9%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      distribute-rgt-out-- [=>] 99.8%	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      associate--l- [=>] 99.8%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      associate--l- [<=] 99.8%	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      sub-neg [=>] 99.8%	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      associate--l+ [=>] 99.8%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      fma-udef [<=] 99.8%	\[ x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, \left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)}\right) \]
      associate-*r* [<=] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 - 2\right)\right)}\right) \]
      metadata-eval [=>] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \]

   3. Applied egg-rr 99.8%

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

      [Start] 99.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot -3\right)\right) \]
      fma-udef [=>] 99.8%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)} \]

   if 7.59999999999999933e153 < x.im

   1. Initial program 53.7%

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

   2. Simplified 66.8%

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

      [Start] 53.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im \]
      *-commutative [<=] 53.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + \color{blue}{x.re \cdot x.im}\right) \cdot x.im \]
      distribute-lft-out [=>] 53.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(x.re \cdot \left(x.im + x.im\right)\right)} \cdot x.im \]
      associate-*l* [=>] 53.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{x.re \cdot \left(\left(x.im + x.im\right) \cdot x.im\right)} \]
      *-commutative [=>] 53.7%	\[ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{\left(\left(x.im + x.im\right) \cdot x.im\right) \cdot x.re} \]
      distribute-rgt-out-- [=>] 53.7%	\[ \color{blue}{x.re \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      associate--l- [=>] 53.7%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re - \left(x.im \cdot x.im + \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      associate--l- [<=] 53.7%	\[ x.re \cdot \color{blue}{\left(\left(x.re \cdot x.re - x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      sub-neg [=>] 53.7%	\[ x.re \cdot \left(\color{blue}{\left(x.re \cdot x.re + \left(-x.im \cdot x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      associate--l+ [=>] 53.7%	\[ x.re \cdot \color{blue}{\left(x.re \cdot x.re + \left(\left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)} \]
      fma-udef [<=] 66.8%	\[ x.re \cdot \color{blue}{\mathsf{fma}\left(x.re, x.re, \left(-x.im \cdot x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \]
      neg-mul-1 [=>] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{-1 \cdot \left(x.im \cdot x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \]
      count-2 [=>] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{\left(2 \cdot x.im\right)} \cdot x.im\right) \]
      associate-*l* [=>] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, -1 \cdot \left(x.im \cdot x.im\right) - \color{blue}{2 \cdot \left(x.im \cdot x.im\right)}\right) \]
      distribute-rgt-out-- [=>] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{\left(x.im \cdot x.im\right) \cdot \left(-1 - 2\right)}\right) \]
      associate-*r* [<=] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot \left(x.im \cdot \left(-1 - 2\right)\right)}\right) \]
      metadata-eval [=>] 66.8%	\[ x.re \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot \left(x.im \cdot \color{blue}{-3}\right)\right) \]

   3. Taylor expanded in x.re around 0 66.8%

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

   4. Simplified 66.8%

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

      [Start] 66.8%	\[ -3 \cdot \left(x.re \cdot {x.im}^{2}\right) \]
      associate-*r* [=>] 66.8%	\[ \color{blue}{\left(-3 \cdot x.re\right) \cdot {x.im}^{2}} \]
      *-commutative [=>] 66.8%	\[ \color{blue}{{x.im}^{2} \cdot \left(-3 \cdot x.re\right)} \]
      *-commutative [=>] 66.8%	\[ {x.im}^{2} \cdot \color{blue}{\left(x.re \cdot -3\right)} \]
      metadata-eval [<=] 66.8%	\[ {x.im}^{2} \cdot \left(x.re \cdot \color{blue}{\left(-2 + -1\right)}\right) \]
      distribute-rgt-out [<=] 66.8%	\[ {x.im}^{2} \cdot \color{blue}{\left(-2 \cdot x.re + -1 \cdot x.re\right)} \]
      distribute-lft-in [=>] 66.8%	\[ \color{blue}{{x.im}^{2} \cdot \left(-2 \cdot x.re\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right)} \]
      metadata-eval [<=] 66.8%	\[ {x.im}^{2} \cdot \left(\color{blue}{\left(-2\right)} \cdot x.re\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-neg-in [<=] 66.8%	\[ {x.im}^{2} \cdot \color{blue}{\left(-2 \cdot x.re\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      count-2 [<=] 66.8%	\[ {x.im}^{2} \cdot \left(-\color{blue}{\left(x.re + x.re\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-neg-in [<=] 66.8%	\[ \color{blue}{\left(-{x.im}^{2} \cdot \left(x.re + x.re\right)\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-out [<=] 66.8%	\[ \left(-\color{blue}{\left(x.re \cdot {x.im}^{2} + x.re \cdot {x.im}^{2}\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-out [=>] 66.8%	\[ \left(-\color{blue}{x.re \cdot \left({x.im}^{2} + {x.im}^{2}\right)}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-rgt-neg-in [=>] 66.8%	\[ \color{blue}{x.re \cdot \left(-\left({x.im}^{2} + {x.im}^{2}\right)\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      count-2 [=>] 66.8%	\[ x.re \cdot \left(-\color{blue}{2 \cdot {x.im}^{2}}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      distribute-lft-neg-in [=>] 66.8%	\[ x.re \cdot \color{blue}{\left(\left(-2\right) \cdot {x.im}^{2}\right)} + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      metadata-eval [=>] 66.8%	\[ x.re \cdot \left(\color{blue}{-2} \cdot {x.im}^{2}\right) + {x.im}^{2} \cdot \left(-1 \cdot x.re\right) \]
      mul-1-neg [=>] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + {x.im}^{2} \cdot \color{blue}{\left(-x.re\right)} \]
      distribute-rgt-neg-in [<=] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(-{x.im}^{2} \cdot x.re\right)} \]
      distribute-lft-neg-in [=>] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(-{x.im}^{2}\right) \cdot x.re} \]
      unpow2 [=>] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \left(-\color{blue}{x.im \cdot x.im}\right) \cdot x.re \]
      distribute-rgt-neg-out [<=] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{\left(x.im \cdot \left(-x.im\right)\right)} \cdot x.re \]
      *-commutative [<=] 66.8%	\[ x.re \cdot \left(-2 \cdot {x.im}^{2}\right) + \color{blue}{x.re \cdot \left(x.im \cdot \left(-x.im\right)\right)} \]

   5. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{{\left(\sqrt{x.re \cdot -3} \cdot x.im\right)}^{2}} \]
      Step-by-step derivation

      [Start] 66.8%	\[ x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right) \]
      add-sqr-sqrt [=>] 22.5%	\[ \color{blue}{\sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)} \cdot \sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)}} \]
      pow2 [=>] 22.5%	\[ \color{blue}{{\left(\sqrt{x.re \cdot \left(-3 \cdot \left(x.im \cdot x.im\right)\right)}\right)}^{2}} \]
      associate-*r* [=>] 22.5%	\[ {\left(\sqrt{\color{blue}{\left(x.re \cdot -3\right) \cdot \left(x.im \cdot x.im\right)}}\right)}^{2} \]
      sqrt-prod [=>] 22.5%	\[ {\color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.im \cdot x.im}\right)}}^{2} \]
      sqrt-prod [=>] 34.7%	\[ {\left(\sqrt{x.re \cdot -3} \cdot \color{blue}{\left(\sqrt{x.im} \cdot \sqrt{x.im}\right)}\right)}^{2} \]
      add-sqr-sqrt [<=] 34.7%	\[ {\left(\sqrt{x.re \cdot -3} \cdot \color{blue}{x.im}\right)}^{2} \]

   6. Applied egg-rr 87.0%

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

      [Start] 34.7%	\[ {\left(\sqrt{x.re \cdot -3} \cdot x.im\right)}^{2} \]
      unpow2 [=>] 34.7%	\[ \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot x.im\right) \cdot \left(\sqrt{x.re \cdot -3} \cdot x.im\right)} \]
      swap-sqr [=>] 22.5%	\[ \color{blue}{\left(\sqrt{x.re \cdot -3} \cdot \sqrt{x.re \cdot -3}\right) \cdot \left(x.im \cdot x.im\right)} \]
      add-sqr-sqrt [<=] 66.8%	\[ \color{blue}{\left(x.re \cdot -3\right)} \cdot \left(x.im \cdot x.im\right) \]
      associate-*r* [=>] 86.8%	\[ \color{blue}{\left(\left(x.re \cdot -3\right) \cdot x.im\right) \cdot x.im} \]
      *-commutative [=>] 86.8%	\[ \left(\color{blue}{\left(-3 \cdot x.re\right)} \cdot x.im\right) \cdot x.im \]
      associate-*r* [<=] 87.0%	\[ \color{blue}{\left(-3 \cdot \left(x.re \cdot x.im\right)\right)} \cdot x.im \]
      *-commutative [=>] 87.0%	\[ \left(-3 \cdot \color{blue}{\left(x.im \cdot x.re\right)}\right) \cdot x.im \]

3. Recombined 3 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.05 \cdot 10^{+133}:\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re + x.im \cdot \left(x.im \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;-3 \cdot \left(x.re \cdot \left(x.im \cdot x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 3
Accuracy	80.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{+109} \lor \neg \left(x.im \leq 5.2 \cdot 10^{-58}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.im \cdot x.re\right) \cdot -3\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 4
Accuracy	80.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x.im \leq -2 \cdot 10^{+109} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-57}\right):\\ \;\;\;\;\left(x.im \cdot \left(x.im \cdot x.re\right)\right) \cdot -3\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 5
Accuracy	59.1%
Cost	320

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

Reproduce

herbie shell --seed 2023178 
(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)))