math.cube on complex, imaginary part

Percentage Accurate: 82.1% → 99.6%

Time: 7.3s

Alternatives: 12

Speedup: 1.5×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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)))

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);
}

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

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);
}

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)

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

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

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]

Initial Program: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \end{array} \]

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)))

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);
}

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

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);
}

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)

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

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

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]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5e+102) (not (<= x.im 1e+60)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (- (* (* x.re 3.0) (* x.im x.re)) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 1e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = ((x_46_re * 3.0) * (x_46_im * x_46_re)) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-5d+102)) .or. (.not. (x_46im <= 1d+60))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = ((x_46re * 3.0d0) * (x_46im * x_46re)) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 1e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = ((x_46_re * 3.0) * (x_46_im * x_46_re)) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5e+102) or not (x_46_im <= 1e+60):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = ((x_46_re * 3.0) * (x_46_im * x_46_re)) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5e+102) || !(x_46_im <= 1e+60))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(Float64(Float64(x_46_re * 3.0) * Float64(x_46_im * x_46_re)) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -5e+102) || ~((x_46_im <= 1e+60)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = ((x_46_re * 3.0) * (x_46_im * x_46_re)) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5e+102], N[Not[LessEqual[x$46$im, 1e+60]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5e102 or 9.9999999999999995e59 < x.im

   1. Initial program 70.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 78.3%

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

      9. distribute-lft-in 78.3%

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

   3. Applied egg-rr 78.3%

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

      1. +-commutative 78.3%

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

      2. add-cube-cbrt 78.3%

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

      3. fma-def 78.3%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 84.8%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -5e102 < x.im < 9.9999999999999995e59

   1. Initial program 88.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.4%

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

      2. *-commutative 88.4%

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

      3. sub-neg 88.4%

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

      4. distribute-lft-in 88.4%

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

      5. associate-+r+ 88.4%

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

      6. distribute-rgt-neg-out 88.4%

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

      7. unsub-neg 88.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

   6. Applied egg-rr 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 2 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5e+102) (not (<= x.im 2e+60)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (- (* x.re (* 3.0 (* x.im x.re))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-5d+102)) .or. (.not. (x_46im <= 2d+60))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = (x_46re * (3.0d0 * (x_46im * x_46re))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5e+102) or not (x_46_im <= 2e+60):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2e+60))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -5e+102) || ~((x_46_im <= 2e+60)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5e+102], N[Not[LessEqual[x$46$im, 2e+60]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5e102 or 1.9999999999999999e60 < x.im

   1. Initial program 70.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 78.3%

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

      9. distribute-lft-in 78.3%

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

   3. Applied egg-rr 78.3%

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

      1. +-commutative 78.3%

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

      2. add-cube-cbrt 78.3%

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

      3. fma-def 78.3%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 84.8%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -5e102 < x.im < 1.9999999999999999e60

   1. Initial program 88.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.4%

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

      2. *-commutative 88.4%

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

      3. sub-neg 88.4%

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

      4. distribute-lft-in 88.4%

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

      5. associate-+r+ 88.4%

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

      6. distribute-rgt-neg-out 88.4%

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

      7. unsub-neg 88.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 2 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 2.1 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5e+102) (not (<= x.im 2.1e+60)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2.1e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-5d+102)) .or. (.not. (x_46im <= 2.1d+60))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2.1e+60)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5e+102) or not (x_46_im <= 2.1e+60):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5e+102) || !(x_46_im <= 2.1e+60))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -5e+102) || ~((x_46_im <= 2.1e+60)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5e+102], N[Not[LessEqual[x$46$im, 2.1e+60]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5e102 or 2.1000000000000001e60 < x.im

   1. Initial program 70.7%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. *-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 78.3%

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

      9. distribute-lft-in 78.3%

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

   3. Applied egg-rr 78.3%

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

      1. +-commutative 78.3%

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

      2. add-cube-cbrt 78.3%

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

      3. fma-def 78.3%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 84.8%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -5e102 < x.im < 2.1000000000000001e60

   1. Initial program 88.4%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.4%

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

      2. *-commutative 88.4%

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

      3. sub-neg 88.4%

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

      4. distribute-lft-in 88.4%

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

      5. associate-+r+ 88.4%

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

      6. distribute-rgt-neg-out 88.4%

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

      7. unsub-neg 88.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+102} \lor \neg \left(x.im \leq 2.1 \cdot 10^{+60}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 4: 91.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7.4e+146)
   (* x.im (- (* 3.0 (* x.re x.re)) (* x.im x.im)))
   (* (* x.re 3.0) (* x.im x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.4e+146) {
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	}
	return tmp;
}

Fortran

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 <= 7.4d+146) then
        tmp = x_46im * ((3.0d0 * (x_46re * x_46re)) - (x_46im * x_46im))
    else
        tmp = (x_46re * 3.0d0) * (x_46im * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7.4e+146) {
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7.4e+146:
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im))
	else:
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7.4e+146)
		tmp = Float64(x_46_im * Float64(Float64(3.0 * Float64(x_46_re * x_46_re)) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_im * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 7.4e+146)
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	else
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7.4e+146], N[(x$46$im * N[(N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7.40000000000000009e146

   1. Initial program 87.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 87.3%

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

      2. *-commutative 87.3%

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

      3. sub-neg 87.3%

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

      4. distribute-lft-in 85.0%

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

      5. associate-+r+ 85.0%

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

      6. distribute-rgt-neg-out 85.0%

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

      7. unsub-neg 85.0%

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

      8. associate-*r* 88.4%

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

      9. distribute-rgt-out 88.4%

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

      10. *-commutative 88.4%

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

      11. count-2 88.4%

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

      12. distribute-lft1-in 88.4%

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

      13. metadata-eval 88.4%

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

      14. *-commutative 88.4%

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

      15. *-commutative 88.4%

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

      16. associate-*r* 88.4%

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

      17. cube-unmult 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x.re around 0 88.4%

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

      1. sub-neg 88.4%

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

      2. *-commutative 88.4%

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

      3. *-commutative 88.4%

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

      4. associate-*l* 88.5%

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

   6. Applied egg-rr 88.5%

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

   8. Simplified 88.5%

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

      1. associate-*l* 85.1%

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

      2. cube-mult 85.0%

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

      3. distribute-lft-out-- 93.2%

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

      4. *-commutative 93.2%

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

      5. associate-*r* 93.2%

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

   10. Applied egg-rr 93.2%

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

   if 7.40000000000000009e146 < x.re

   1. Initial program 51.0%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 51.0%

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

      2. *-commutative 51.0%

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

      3. sub-neg 51.0%

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

      4. distribute-lft-in 42.9%

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

      5. associate-+r+ 42.9%

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

      6. distribute-rgt-neg-out 42.9%

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

      7. unsub-neg 42.9%

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

      8. associate-*r* 72.8%

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

      9. distribute-rgt-out 72.9%

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

      10. *-commutative 72.9%

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

      11. count-2 72.9%

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

      12. distribute-lft1-in 72.9%

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

      13. metadata-eval 72.9%

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

      14. *-commutative 72.9%

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

      15. *-commutative 72.9%

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

      16. associate-*r* 72.9%

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

      17. cube-unmult 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x.re around 0 72.9%

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

      1. sub-neg 72.9%

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

      2. *-commutative 72.9%

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

      3. *-commutative 72.9%

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

      4. associate-*l* 72.8%

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

   6. Applied egg-rr 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in x.im around 0 64.5%

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

       1. unpow2 64.5%

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

       2. associate-*r* 64.5%

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

       3. *-commutative 64.5%

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

       4. associate-*l* 64.5%

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

       5. associate-*r* 94.6%

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

       6. *-commutative 94.6%

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

       7. associate-*l* 94.5%

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

   11. Simplified 94.5%

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

       1. expm1-log1p-u 44.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)\right)} \]

       2. expm1-udef 44.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - 1} \]

       3. *-commutative 44.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re}\right)} - 1 \]

       4. *-commutative 44.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 3\right)} \cdot x.re\right)} - 1 \]

       5. associate-*l* 44.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)}\right)} - 1 \]

   13. Applied egg-rr 44.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)\right)} - 1} \]

   15. Simplified 94.5%

       \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(3 \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7.4 \cdot 10^{+146}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]

Alternative 5: 67.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.4e+23)
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.4e+23) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

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 <= 1.4d+23) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.4e+23) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.4e+23:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.4e+23)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.4e+23)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.4e+23], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.4e23

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. distribute-lft-in 85.7%

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

      5. associate-+r+ 85.8%

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

      6. distribute-rgt-neg-out 85.8%

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

      7. unsub-neg 85.8%

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

      8. associate-*r* 89.6%

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

      9. distribute-rgt-out 89.5%

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

      10. *-commutative 89.5%

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

      11. count-2 89.5%

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

      12. distribute-lft1-in 89.5%

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

      13. metadata-eval 89.5%

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

      14. *-commutative 89.5%

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

      15. *-commutative 89.5%

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

      16. associate-*r* 89.6%

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

      17. cube-unmult 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x.re around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*l* 89.6%

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

   6. Applied egg-rr 89.6%

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

   8. Simplified 89.6%

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

      1. associate-*l* 85.8%

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

      2. cube-mult 85.7%

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

      3. distribute-lft-out-- 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   10. Applied egg-rr 92.4%

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

   11. Taylor expanded in x.re around 0 70.4%

       \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.4%

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

       2. mul-1-neg 70.4%

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

       3. distribute-rgt-neg-out 70.4%

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

   13. Simplified 70.4%

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

   if 1.4e23 < x.re

   1. Initial program 61.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

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

      2. *-commutative 61.9%

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

      3. sub-neg 61.9%

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

      4. distribute-lft-in 57.0%

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

      5. associate-+r+ 57.0%

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

      6. distribute-rgt-neg-out 57.0%

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

      7. unsub-neg 57.0%

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

      8. associate-*r* 75.2%

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

      9. distribute-rgt-out 75.2%

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

      10. *-commutative 75.2%

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

      11. count-2 75.2%

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

      12. distribute-lft1-in 75.2%

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

      13. metadata-eval 75.2%

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

      14. *-commutative 75.2%

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

      15. *-commutative 75.2%

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

      16. associate-*r* 75.2%

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

      17. cube-unmult 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x.re around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*l* 75.2%

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

   6. Applied egg-rr 75.2%

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

   8. Simplified 75.2%

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

   9. Taylor expanded in x.im around 0 68.5%

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

       1. unpow2 68.5%

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

       2. *-commutative 68.5%

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

   11. Simplified 68.5%

       \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 6: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7 \cdot 10^{+21}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7e+21)
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.re (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7e+21) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

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 <= 7d+21) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46re * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 7e+21) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 7e+21:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 7e+21)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 7e+21)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 7e+21], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 7e21

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. distribute-lft-in 85.7%

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

      5. associate-+r+ 85.8%

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

      6. distribute-rgt-neg-out 85.8%

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

      7. unsub-neg 85.8%

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

      8. associate-*r* 89.6%

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

      9. distribute-rgt-out 89.5%

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

      10. *-commutative 89.5%

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

      11. count-2 89.5%

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

      12. distribute-lft1-in 89.5%

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

      13. metadata-eval 89.5%

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

      14. *-commutative 89.5%

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

      15. *-commutative 89.5%

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

      16. associate-*r* 89.6%

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

      17. cube-unmult 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x.re around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*l* 89.6%

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

   6. Applied egg-rr 89.6%

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

   8. Simplified 89.6%

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

      1. associate-*l* 85.8%

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

      2. cube-mult 85.7%

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

      3. distribute-lft-out-- 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   10. Applied egg-rr 92.4%

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

   11. Taylor expanded in x.re around 0 70.4%

       \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.4%

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

       2. mul-1-neg 70.4%

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

       3. distribute-rgt-neg-out 70.4%

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

   13. Simplified 70.4%

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

   if 7e21 < x.re

   1. Initial program 61.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

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

      2. *-commutative 61.9%

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

      3. sub-neg 61.9%

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

      4. distribute-lft-in 57.0%

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

      5. associate-+r+ 57.0%

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

      6. distribute-rgt-neg-out 57.0%

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

      7. unsub-neg 57.0%

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

      8. associate-*r* 75.2%

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

      9. distribute-rgt-out 75.2%

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

      10. *-commutative 75.2%

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

      11. count-2 75.2%

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

      12. distribute-lft1-in 75.2%

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

      13. metadata-eval 75.2%

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

      14. *-commutative 75.2%

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

      15. *-commutative 75.2%

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

      16. associate-*r* 75.2%

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

      17. cube-unmult 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x.re around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*l* 75.2%

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

   6. Applied egg-rr 75.2%

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

   8. Simplified 75.2%

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

      1. associate-*l* 57.0%

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

      2. cube-mult 57.0%

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

      3. distribute-lft-out-- 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r* 70.1%

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

   10. Applied egg-rr 70.1%

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

   11. Taylor expanded in x.im around 0 68.5%

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

       1. unpow2 68.5%

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

       2. *-commutative 68.5%

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

       3. associate-*r* 86.6%

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

       4. *-commutative 86.6%

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

       5. *-commutative 86.6%

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

   13. Simplified 86.6%

       \[\leadsto \color{blue}{3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7 \cdot 10^{+21}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 7: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.22e+23)
   (* x.im (* x.im (- x.im)))
   (* x.re (* 3.0 (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.22e+23) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

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 <= 1.22d+23) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * (3.0d0 * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.22e+23) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.22e+23:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.22e+23)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.22e+23)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.22e+23], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.22e23

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. distribute-lft-in 85.7%

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

      5. associate-+r+ 85.8%

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

      6. distribute-rgt-neg-out 85.8%

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

      7. unsub-neg 85.8%

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

      8. associate-*r* 89.6%

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

      9. distribute-rgt-out 89.5%

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

      10. *-commutative 89.5%

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

      11. count-2 89.5%

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

      12. distribute-lft1-in 89.5%

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

      13. metadata-eval 89.5%

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

      14. *-commutative 89.5%

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

      15. *-commutative 89.5%

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

      16. associate-*r* 89.6%

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

      17. cube-unmult 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x.re around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*l* 89.6%

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

   6. Applied egg-rr 89.6%

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

   8. Simplified 89.6%

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

      1. associate-*l* 85.8%

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

      2. cube-mult 85.7%

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

      3. distribute-lft-out-- 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   10. Applied egg-rr 92.4%

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

   11. Taylor expanded in x.re around 0 70.4%

       \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.4%

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

       2. mul-1-neg 70.4%

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

       3. distribute-rgt-neg-out 70.4%

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

   13. Simplified 70.4%

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

   if 1.22e23 < x.re

   1. Initial program 61.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

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

      2. *-commutative 61.9%

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

      3. sub-neg 61.9%

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

      4. distribute-lft-in 57.0%

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

      5. associate-+r+ 57.0%

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

      6. distribute-rgt-neg-out 57.0%

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

      7. unsub-neg 57.0%

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

      8. associate-*r* 75.2%

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

      9. distribute-rgt-out 75.2%

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

      10. *-commutative 75.2%

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

      11. count-2 75.2%

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

      12. distribute-lft1-in 75.2%

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

      13. metadata-eval 75.2%

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

      14. *-commutative 75.2%

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

      15. *-commutative 75.2%

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

      16. associate-*r* 75.2%

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

      17. cube-unmult 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x.re around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*l* 75.2%

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

   6. Applied egg-rr 75.2%

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

   8. Simplified 75.2%

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

   9. Taylor expanded in x.im around 0 68.5%

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

       1. unpow2 68.5%

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

       2. associate-*r* 68.5%

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

       3. *-commutative 68.5%

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

       4. associate-*l* 68.5%

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

       5. associate-*r* 86.7%

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

       6. *-commutative 86.7%

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

       7. associate-*l* 86.7%

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

   11. Simplified 86.7%

       \[\leadsto \color{blue}{x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{+22}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 3.4e+22)
   (* x.im (* x.im (- x.im)))
   (* (* x.re 3.0) (* x.im x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 3.4e+22) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	}
	return tmp;
}

Fortran

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 <= 3.4d+22) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = (x_46re * 3.0d0) * (x_46im * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 3.4e+22) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 3.4e+22:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 3.4e+22)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * 3.0) * Float64(x_46_im * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 3.4e+22)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = (x_46_re * 3.0) * (x_46_im * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 3.4e+22], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.4e22

   1. Initial program 88.3%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. sub-neg 88.3%

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

      4. distribute-lft-in 85.7%

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

      5. associate-+r+ 85.8%

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

      6. distribute-rgt-neg-out 85.8%

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

      7. unsub-neg 85.8%

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

      8. associate-*r* 89.6%

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

      9. distribute-rgt-out 89.5%

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

      10. *-commutative 89.5%

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

      11. count-2 89.5%

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

      12. distribute-lft1-in 89.5%

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

      13. metadata-eval 89.5%

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

      14. *-commutative 89.5%

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

      15. *-commutative 89.5%

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

      16. associate-*r* 89.6%

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

      17. cube-unmult 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x.re around 0 89.6%

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

      1. sub-neg 89.6%

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

      2. *-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*l* 89.6%

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

   6. Applied egg-rr 89.6%

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

   8. Simplified 89.6%

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

      1. associate-*l* 85.8%

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

      2. cube-mult 85.7%

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

      3. distribute-lft-out-- 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   10. Applied egg-rr 92.4%

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

   11. Taylor expanded in x.re around 0 70.4%

       \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 70.4%

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

       2. mul-1-neg 70.4%

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

       3. distribute-rgt-neg-out 70.4%

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

   13. Simplified 70.4%

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

   if 3.4e22 < x.re

   1. Initial program 61.9%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
   2. Step-by-step derivation

      1. +-commutative 61.9%

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

      2. *-commutative 61.9%

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

      3. sub-neg 61.9%

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

      4. distribute-lft-in 57.0%

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

      5. associate-+r+ 57.0%

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

      6. distribute-rgt-neg-out 57.0%

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

      7. unsub-neg 57.0%

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

      8. associate-*r* 75.2%

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

      9. distribute-rgt-out 75.2%

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

      10. *-commutative 75.2%

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

      11. count-2 75.2%

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

      12. distribute-lft1-in 75.2%

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

      13. metadata-eval 75.2%

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

      14. *-commutative 75.2%

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

      15. *-commutative 75.2%

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

      16. associate-*r* 75.2%

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

      17. cube-unmult 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x.re around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*l* 75.2%

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

   6. Applied egg-rr 75.2%

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

   8. Simplified 75.2%

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

   9. Taylor expanded in x.im around 0 68.5%

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

       1. unpow2 68.5%

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

       2. associate-*r* 68.5%

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

       3. *-commutative 68.5%

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

       4. associate-*l* 68.5%

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

       5. associate-*r* 86.7%

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

       6. *-commutative 86.7%

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

       7. associate-*l* 86.7%

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

   11. Simplified 86.7%

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

       1. expm1-log1p-u 50.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)\right)} \]

       2. expm1-udef 41.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.re \cdot \left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - 1} \]

       3. *-commutative 41.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.re}\right)} - 1 \]

       4. *-commutative 41.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(x.re \cdot x.im\right) \cdot 3\right)} \cdot x.re\right)} - 1 \]

       5. associate-*l* 41.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)}\right)} - 1 \]

   13. Applied egg-rr 41.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(x.re \cdot x.im\right) \cdot \left(3 \cdot x.re\right)\right)} - 1} \]

   15. Simplified 86.7%

       \[\leadsto \color{blue}{\left(x.im \cdot x.re\right) \cdot \left(3 \cdot x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.4 \cdot 10^{+22}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]

Alternative 9: 59.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(x.im \cdot \left(-x.im\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* x.im (- x.im))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = x_46im * (x_46im * -x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_im * -x_46_im);
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_im * -x_46_im)

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_im * -x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 82.0%

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

   2. *-commutative 82.0%

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

   3. sub-neg 82.0%

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

   4. distribute-lft-in 78.9%

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

   5. associate-+r+ 78.9%

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

   6. distribute-rgt-neg-out 78.9%

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

   7. unsub-neg 78.9%

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

   8. associate-*r* 86.1%

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

   9. distribute-rgt-out 86.1%

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

   10. *-commutative 86.1%

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

   11. count-2 86.1%

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

   12. distribute-lft1-in 86.1%

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

   13. metadata-eval 86.1%

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

   14. *-commutative 86.1%

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

   15. *-commutative 86.1%

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

   16. associate-*r* 86.1%

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

   17. cube-unmult 86.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in x.re around 0 86.2%

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

   1. sub-neg 86.2%

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

   2. *-commutative 86.2%

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

   3. *-commutative 86.2%

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

   4. associate-*l* 86.2%

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

6. Applied egg-rr 86.2%

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

8. Simplified 86.2%

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

   1. associate-*l* 79.0%

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

   2. cube-mult 78.9%

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

   3. distribute-lft-out-- 87.1%

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

   4. *-commutative 87.1%

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

   5. associate-*r* 87.1%

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

10. Applied egg-rr 87.1%

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

11. Taylor expanded in x.re around 0 57.0%

    \[\leadsto x.im \cdot \color{blue}{\left(-1 \cdot {x.im}^{2}\right)} \]
12. Step-by-step derivation

    1. unpow2 57.0%

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

    2. mul-1-neg 57.0%

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

    3. distribute-rgt-neg-out 57.0%

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

13. Simplified 57.0%

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

14. Final simplification 57.0%

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

Alternative 10: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -10 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -10.0)

C

double code(double x_46_re, double x_46_im) {
	return -10.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -10.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -10.0;
}

Python

def code(x_46_re, x_46_im):
	return -10.0

Julia

function code(x_46_re, x_46_im)
	return -10.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -10.0;
end

Wolfram

code[x$46$re_, x$46$im_] := -10.0

Derivation

1. Initial program 82.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 82.0%

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

   2. *-commutative 82.0%

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

   3. sub-neg 82.0%

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

   4. distribute-lft-in 78.9%

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

   5. associate-+r+ 78.9%

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

   6. distribute-rgt-neg-out 78.9%

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

   7. unsub-neg 78.9%

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

   8. associate-*r* 86.1%

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

   9. distribute-rgt-out 86.1%

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

   10. *-commutative 86.1%

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

   11. count-2 86.1%

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

   12. distribute-lft1-in 86.1%

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

   13. metadata-eval 86.1%

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

   14. *-commutative 86.1%

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

   15. *-commutative 86.1%

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

   16. associate-*r* 86.1%

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

   17. cube-unmult 86.2%

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

3. Simplified 86.2%

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

   1. associate-*r* 86.2%

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

   2. associate-*l* 86.2%

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

   3. flip-- 22.7%

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

   4. div-inv 22.3%

      \[\leadsto \color{blue}{\left(\left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) \cdot \left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}}} \]

   5. swap-sqr 22.3%

      \[\leadsto \left(\color{blue}{\left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\right) \cdot \left(3 \cdot 3\right)} - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   6. pow2 22.3%

      \[\leadsto \left(\color{blue}{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2}} \cdot \left(3 \cdot 3\right) - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   7. metadata-eval 22.3%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot \color{blue}{9} - {x.im}^{3} \cdot {x.im}^{3}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   8. pow-prod-up 22.3%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - \color{blue}{{x.im}^{\left(3 + 3\right)}}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   9. metadata-eval 22.3%

      \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{\color{blue}{6}}\right) \cdot \frac{1}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   10. associate-*l* 22.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)} + {x.im}^{3}} \]

   11. associate-*r* 22.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} + {x.im}^{3}} \]

   12. fma-def 22.2%

       \[\leadsto \left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

5. Applied egg-rr 22.2%

   \[\leadsto \color{blue}{\left({\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}\right) \cdot \frac{1}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

7. Simplified 2.6%

   \[\leadsto \color{blue}{-10} \]

8. Final simplification 2.6%

   \[\leadsto -10 \]

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.1 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 0.1)

C

double code(double x_46_re, double x_46_im) {
	return 0.1;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.1d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.1;
}

Python

def code(x_46_re, x_46_im):
	return 0.1

Julia

function code(x_46_re, x_46_im)
	return 0.1
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.1;
end

Wolfram

code[x$46$re_, x$46$im_] := 0.1

Derivation

1. Initial program 82.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 82.0%

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

   2. *-commutative 82.0%

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

   3. sub-neg 82.0%

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

   4. distribute-lft-in 78.9%

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

   5. associate-+r+ 78.9%

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

   6. distribute-rgt-neg-out 78.9%

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

   7. unsub-neg 78.9%

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

   8. associate-*r* 86.1%

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

   9. distribute-rgt-out 86.1%

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

   10. *-commutative 86.1%

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

   11. count-2 86.1%

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

   12. distribute-lft1-in 86.1%

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

   13. metadata-eval 86.1%

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

   14. *-commutative 86.1%

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

   15. *-commutative 86.1%

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

   16. associate-*r* 86.1%

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

   17. cube-unmult 86.2%

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

3. Simplified 86.2%

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

   1. sub-neg 86.2%

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

   2. associate-*r* 86.2%

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

   3. associate-*l* 86.2%

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

   4. flip3-+ 13.5%

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

   5. associate-*r* 13.1%

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

   6. associate-*r* 13.0%

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

   7. unpow-prod-down 7.6%

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

   8. pow2 7.6%

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

   9. pow-pow 7.7%

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

   10. metadata-eval 7.7%

       \[\leadsto \frac{{x.re}^{\color{blue}{6}} \cdot {\left(x.im \cdot 3\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{\left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) \cdot \left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3\right) \cdot \left(-{x.im}^{3}\right)\right)} \]

5. Applied egg-rr 7.7%

   \[\leadsto \color{blue}{\frac{{x.re}^{6} \cdot {\left(x.im \cdot 3\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\right) \cdot \left(-{x.im}^{3}\right)\right)}} \]

7. Simplified 2.8%

   \[\leadsto \color{blue}{0.1} \]

8. Final simplification 2.8%

   \[\leadsto 0.1 \]

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 10 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 10.0)

C

double code(double x_46_re, double x_46_im) {
	return 10.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 10.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 10.0;
}

Python

def code(x_46_re, x_46_im):
	return 10.0

Julia

function code(x_46_re, x_46_im)
	return 10.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 10.0;
end

Wolfram

code[x$46$re_, x$46$im_] := 10.0

Derivation

1. Initial program 82.0%

   \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
2. Step-by-step derivation

   1. +-commutative 82.0%

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

   2. *-commutative 82.0%

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

   3. sub-neg 82.0%

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

   4. distribute-lft-in 78.9%

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

   5. associate-+r+ 78.9%

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

   6. distribute-rgt-neg-out 78.9%

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

   7. unsub-neg 78.9%

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

   8. associate-*r* 86.1%

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

   9. distribute-rgt-out 86.1%

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

   10. *-commutative 86.1%

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

   11. count-2 86.1%

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

   12. distribute-lft1-in 86.1%

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

   13. metadata-eval 86.1%

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

   14. *-commutative 86.1%

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

   15. *-commutative 86.1%

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

   16. associate-*r* 86.1%

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

   17. cube-unmult 86.2%

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

3. Simplified 86.2%

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

   1. associate-*r* 86.2%

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

   2. associate-*l* 86.2%

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

   3. flip-- 22.7%

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

   4. swap-sqr 22.7%

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

   5. pow2 22.7%

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

   6. metadata-eval 22.7%

      \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot \color{blue}{9} - {x.im}^{3} \cdot {x.im}^{3}}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   7. pow-prod-up 22.6%

      \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - \color{blue}{{x.im}^{\left(3 + 3\right)}}}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   8. metadata-eval 22.6%

      \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{\color{blue}{6}}}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right) \cdot 3 + {x.im}^{3}} \]

   9. associate-*l* 22.6%

      \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}}{\color{blue}{x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)} + {x.im}^{3}} \]

   10. associate-*r* 22.6%

       \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}}{x.re \cdot \color{blue}{\left(x.re \cdot \left(x.im \cdot 3\right)\right)} + {x.im}^{3}} \]

   11. fma-def 22.6%

       \[\leadsto \frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}}{\color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

5. Applied egg-rr 22.6%

   \[\leadsto \color{blue}{\frac{{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{2} \cdot 9 - {x.im}^{6}}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im \cdot 3\right), {x.im}^{3}\right)}} \]

7. Simplified 2.8%

   \[\leadsto \color{blue}{10} \]

8. Final simplification 2.8%

   \[\leadsto 10 \]

Developer target: 91.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right) \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (+ (* (* x.re x.im) (* 2.0 x.re)) (* (* x.im (- 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_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

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_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
}

Python

def code(x_46_re, x_46_im):
	return ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(Float64(x_46_re * x_46_im) * Float64(2.0 * x_46_re)) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = ((x_46_re * x_46_im) * (2.0 * x_46_re)) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(2.0 * x$46$re), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision] * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023217 
(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)))