math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.8%

Time: 8.2s

Alternatives: 15

Speedup: 1.3×

• 44.1% 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: 82.3% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right)\\ \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;t_0 + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + x.re \cdot 0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (- x.re x.im) (* x.im (+ x.re x.im)))))
   (if (<=
        (+
         (* x.im (- (* x.re x.re) (* x.im x.im)))
         (* x.re (+ (* x.re x.im) (* x.re x.im))))
        INFINITY)
     (+ t_0 (* x.re (* x.re (+ x.im x.im))))
     (+ t_0 (* x.re 0.0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= ((double) INFINITY)) {
		tmp = t_0 + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = t_0 + (x_46_re * 0.0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	} else {
		tmp = t_0 + (x_46_re * 0.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = t_0 + (x_46_re * (x_46_re * (x_46_im + x_46_im)))
	else:
		tmp = t_0 + (x_46_re * 0.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im * Float64(x_46_re + x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(x_46_re * Float64(x_46_im + x_46_im))));
	else
		tmp = Float64(t_0 + Float64(x_46_re * 0.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = t_0 + (x_46_re * (x_46_re * (x_46_im + x_46_im)));
	else
		tmp = t_0 + (x_46_re * 0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 + N[(x$46$re * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(x$46$re * 0.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re)) < +inf.0

   1. Initial program 91.8%

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

      1. add-cube-cbrt 91.4%

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

      2. pow3 91.5%

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

      3. *-commutative 91.5%

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

      4. difference-of-squares 91.5%

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

      5. associate-*r* 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow3 99.4%

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

      2. add-cube-cbrt 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   7. Applied egg-rr 99.7%

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

   if +inf.0 < (+.f64 (*.f64 (-.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)) x.im) (*.f64 (+.f64 (*.f64 x.re x.im) (*.f64 x.im x.re)) x.re))

   1. Initial program 0.0%

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

      1. add-cube-cbrt 0.0%

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

      2. pow3 0.0%

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

      3. *-commutative 0.0%

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

      4. difference-of-squares 11.8%

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

      5. associate-*r* 11.8%

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

   3. Applied egg-rr 11.8%

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

      1. unpow3 11.8%

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

      2. add-cube-cbrt 11.8%

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

      3. *-commutative 11.8%

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

      4. +-commutative 11.8%

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

   5. Applied egg-rr 11.8%

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

      1. *-commutative 11.8%

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

      2. flip-+ 0.0%

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

      3. div-sub 0.0%

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

      4. pow2 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

      11. *-commutative 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. +-inverses 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.im + x.re\right)\right) + \color{blue}{0} \cdot x.re \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq \infty:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \end{array} \]

Alternative 2: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{-105} \lor \neg \left(x.im \leq 3.1 \cdot 10^{-79}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.25e-105) (not (<= x.im 3.1e-79)))
   (+ (* (- x.re x.im) (* x.im (+ x.re x.im))) (* x.re 0.0))
   (* (* x.re x.im) (* x.re 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.25e-105) || !(x_46_im <= 3.1e-79)) {
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + (x_46_re * 0.0);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.25d-105)) .or. (.not. (x_46im <= 3.1d-79))) then
        tmp = ((x_46re - x_46im) * (x_46im * (x_46re + x_46im))) + (x_46re * 0.0d0)
    else
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.25e-105) || !(x_46_im <= 3.1e-79)) {
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + (x_46_re * 0.0);
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.25e-105) or not (x_46_im <= 3.1e-79):
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + (x_46_re * 0.0)
	else:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.25e-105) || !(x_46_im <= 3.1e-79))
		tmp = Float64(Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im * Float64(x_46_re + x_46_im))) + Float64(x_46_re * 0.0));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.25e-105) || ~((x_46_im <= 3.1e-79)))
		tmp = ((x_46_re - x_46_im) * (x_46_im * (x_46_re + x_46_im))) + (x_46_re * 0.0);
	else
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.25e-105], N[Not[LessEqual[x$46$im, 3.1e-79]], $MachinePrecision]], N[(N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.24999999999999991e-105 or 3.0999999999999999e-79 < x.im

   1. Initial program 78.7%

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

      1. add-cube-cbrt 78.3%

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

      2. pow3 78.4%

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

      3. *-commutative 78.4%

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

      4. difference-of-squares 80.9%

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

      5. associate-*r* 80.9%

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

   3. Applied egg-rr 80.9%

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

      1. unpow3 80.8%

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

      2. add-cube-cbrt 81.2%

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

      3. *-commutative 81.2%

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

      4. +-commutative 81.2%

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

   5. Applied egg-rr 81.2%

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

      1. *-commutative 81.2%

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

      2. flip-+ 0.0%

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

      3. div-sub 0.0%

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

      4. pow2 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

      11. *-commutative 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. +-inverses 92.7%

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

   9. Simplified 92.7%

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

   if -1.24999999999999991e-105 < x.im < 3.0999999999999999e-79

   1. Initial program 81.2%

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

      1. add-cube-cbrt 80.8%

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

      2. pow3 80.9%

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

      3. *-commutative 80.9%

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

      4. difference-of-squares 80.9%

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

      5. associate-*r* 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow3 99.4%

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

      2. add-cube-cbrt 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x.re around inf 79.2%

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

      1. distribute-rgt1-in 79.2%

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

      2. metadata-eval 79.2%

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

      3. associate-*r* 79.3%

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

      4. unpow2 79.3%

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

      5. associate-*r* 79.3%

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

      6. *-commutative 79.3%

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

      7. associate-*r* 97.8%

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

      8. *-commutative 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{-105} \lor \neg \left(x.im \leq 3.1 \cdot 10^{-79}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 3: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+39} \lor \neg \left(x.im \leq 7.5 \cdot 10^{-56}\right):\\ \;\;\;\;x.re \cdot 0 + \left(x.im \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.4e+39) (not (<= x.im 7.5e-56)))
   (+ (* x.re 0.0) (* (* x.im x.im) (- x.re x.im)))
   (* (* x.re x.im) (* x.re 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+39) || !(x_46_im <= 7.5e-56)) {
		tmp = (x_46_re * 0.0) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-3.4d+39)) .or. (.not. (x_46im <= 7.5d-56))) then
        tmp = (x_46re * 0.0d0) + ((x_46im * x_46im) * (x_46re - x_46im))
    else
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.4e+39) || !(x_46_im <= 7.5e-56)) {
		tmp = (x_46_re * 0.0) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -3.4e+39) or not (x_46_im <= 7.5e-56):
		tmp = (x_46_re * 0.0) + ((x_46_im * x_46_im) * (x_46_re - x_46_im))
	else:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3.4e+39) || !(x_46_im <= 7.5e-56))
		tmp = Float64(Float64(x_46_re * 0.0) + Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3.4e+39) || ~((x_46_im <= 7.5e-56)))
		tmp = (x_46_re * 0.0) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	else
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3.4e+39], N[Not[LessEqual[x$46$im, 7.5e-56]], $MachinePrecision]], N[(N[(x$46$re * 0.0), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.3999999999999999e39 or 7.50000000000000041e-56 < x.im

   1. Initial program 73.5%

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

      1. add-cube-cbrt 73.2%

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

      2. pow3 73.3%

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

      3. *-commutative 73.3%

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

      4. difference-of-squares 76.4%

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

      5. associate-*r* 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. unpow3 76.3%

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

      2. add-cube-cbrt 76.6%

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

      3. *-commutative 76.6%

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

      4. +-commutative 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. flip-+ 0.0%

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

      3. div-sub 0.0%

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

      4. pow2 0.0%

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

      5. *-commutative 0.0%

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

      6. *-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. pow2 0.0%

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

      9. *-commutative 0.0%

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

      10. *-commutative 0.0%

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

      11. *-commutative 0.0%

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

   7. Applied egg-rr 0.0%

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

      1. +-inverses 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x.im around inf 86.9%

       \[\leadsto \left(x.re - x.im\right) \cdot \color{blue}{{x.im}^{2}} + 0 \cdot x.re \]
   11. Step-by-step derivation

       1. unpow2 86.9%

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

   12. Simplified 86.9%

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

   if -3.3999999999999999e39 < x.im < 7.50000000000000041e-56

   1. Initial program 85.9%

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

      1. add-cube-cbrt 85.4%

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

      2. pow3 85.5%

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

      3. *-commutative 85.5%

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

      4. difference-of-squares 85.5%

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

      5. associate-*r* 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. unpow3 99.3%

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

      2. add-cube-cbrt 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x.re around inf 74.6%

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

      1. distribute-rgt1-in 74.6%

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

      2. metadata-eval 74.6%

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

      3. associate-*r* 74.6%

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

      4. unpow2 74.6%

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

      5. associate-*r* 74.6%

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

      6. *-commutative 74.6%

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

      7. associate-*r* 88.5%

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

      8. *-commutative 88.5%

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

   10. Simplified 88.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{+39} \lor \neg \left(x.im \leq 7.5 \cdot 10^{-56}\right):\\ \;\;\;\;x.re \cdot 0 + \left(x.im \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \end{array} \]

Alternative 4: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.im + x.im\right)\\ \mathbf{if}\;x.im \leq -3.8 \cdot 10^{+130}:\\ \;\;\;\;t_0 + t_0\\ \mathbf{elif}\;x.im \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.im (+ x.im x.im))))
   (if (<= x.im -3.8e+130)
     (+ t_0 t_0)
     (if (<= x.im 8e+149)
       (* (* x.re x.im) (* x.re 3.0))
       (- (+ x.im x.im) (* x.re (* x.im x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im + x_46_im);
	double tmp;
	if (x_46_im <= -3.8e+130) {
		tmp = t_0 + t_0;
	} else if (x_46_im <= 8e+149) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46im * (x_46im + x_46im)
    if (x_46im <= (-3.8d+130)) then
        tmp = t_0 + t_0
    else if (x_46im <= 8d+149) then
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    else
        tmp = (x_46im + x_46im) - (x_46re * (x_46im * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_im * (x_46_im + x_46_im);
	double tmp;
	if (x_46_im <= -3.8e+130) {
		tmp = t_0 + t_0;
	} else if (x_46_im <= 8e+149) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_im * (x_46_im + x_46_im)
	tmp = 0
	if x_46_im <= -3.8e+130:
		tmp = t_0 + t_0
	elif x_46_im <= 8e+149:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	else:
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_im * Float64(x_46_im + x_46_im))
	tmp = 0.0
	if (x_46_im <= -3.8e+130)
		tmp = Float64(t_0 + t_0);
	elseif (x_46_im <= 8e+149)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	else
		tmp = Float64(Float64(x_46_im + x_46_im) - Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_im * (x_46_im + x_46_im);
	tmp = 0.0;
	if (x_46_im <= -3.8e+130)
		tmp = t_0 + t_0;
	elseif (x_46_im <= 8e+149)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	else
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$im * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -3.8e+130], N[(t$95$0 + t$95$0), $MachinePrecision], If[LessEqual[x$46$im, 8e+149], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] - N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.8000000000000002e130

   1. Initial program 58.5%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

      3. *-commutative 58.5%

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

      4. difference-of-squares 61.0%

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

      5. associate-*r* 61.0%

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

   3. Applied egg-rr 61.0%

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

   4. Taylor expanded in x.im around 0 3.0%

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

      1. *-commutative 3.0%

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

      2. unpow2 3.0%

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

      3. associate-*r* 2.9%

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

   6. Simplified 2.9%

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

      1. *-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. associate-*l* 2.9%

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

      4. *-commutative 2.9%

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

      5. count-2 2.9%

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

      6. flip-+ 0.0%

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

      7. *-commutative 0.0%

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

      8. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\color{blue}{0}}{x.re \cdot x.im - x.im \cdot x.re} \]

      9. +-inverses 0.0%

         \[\leadsto x.re \cdot \frac{\color{blue}{x.im \cdot x.re - x.im \cdot x.re}}{x.re \cdot x.im - x.im \cdot x.re} \]

      10. *-commutative 0.0%

          \[\leadsto x.re \cdot \frac{\color{blue}{x.re \cdot x.im} - x.im \cdot x.re}{x.re \cdot x.im - x.im \cdot x.re} \]

      11. associate-*r/ 0.0%

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

      12. *-commutative 0.0%

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

      13. +-inverses 0.0%

          \[\leadsto \frac{x.re \cdot \color{blue}{0}}{x.re \cdot x.im - x.im \cdot x.re} \]

      14. +-inverses 0.0%

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

      15. distribute-lft-out-- 0.0%

          \[\leadsto \frac{\color{blue}{x.re \cdot x.im - x.re \cdot x.im}}{x.re \cdot x.im - x.im \cdot x.re} \]

      16. *-commutative 0.0%

          \[\leadsto \frac{x.re \cdot x.im - \color{blue}{x.im \cdot x.re}}{x.re \cdot x.im - x.im \cdot x.re} \]

   8. Applied egg-rr 86.6%

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

   if -3.8000000000000002e130 < x.im < 8.00000000000000039e149

   1. Initial program 89.7%

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

      1. add-cube-cbrt 89.1%

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

      2. pow3 89.2%

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

      3. *-commutative 89.2%

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

      4. difference-of-squares 89.2%

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

      5. associate-*r* 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. unpow3 98.7%

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

      2. add-cube-cbrt 99.1%

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

      3. *-commutative 99.1%

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

      4. +-commutative 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-lft-out 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x.re around inf 61.8%

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

      1. distribute-rgt1-in 61.8%

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

      2. metadata-eval 61.8%

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

      3. associate-*r* 61.8%

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

      4. unpow2 61.8%

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

      5. associate-*r* 61.9%

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

      6. *-commutative 61.9%

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

      7. associate-*r* 71.4%

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

      8. *-commutative 71.4%

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

   10. Simplified 71.4%

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

   if 8.00000000000000039e149 < x.im

   1. Initial program 46.7%

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

      1. +-commutative 46.7%

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

      2. *-commutative 46.7%

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

      3. fma-def 50.0%

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

      4. *-commutative 50.0%

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

      5. distribute-rgt-out 50.0%

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

      6. *-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. fma-udef 46.7%

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

      2. distribute-lft-in 46.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 56.7%

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

      9. distribute-lft-in 56.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 73.3%

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

      16. difference-of-squares 100.0%

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

      17. associate-*r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 47.2%

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

   7. Taylor expanded in x.re around 0 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*r* 53.3%

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

      3. mul-1-neg 53.3%

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

   9. Simplified 53.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.8 \cdot 10^{+130}:\\ \;\;\;\;x.im \cdot \left(x.im + x.im\right) + x.im \cdot \left(x.im + x.im\right)\\ \mathbf{elif}\;x.im \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]

Alternative 5: 58.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 4.4 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 4.4e+154)
   (* (* x.re x.im) (* x.re 3.0))
   (- (+ x.im x.im) (* x.im (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.4e+154) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 4.4d+154) then
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    else
        tmp = (x_46im + x_46im) - (x_46im * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 4.4e+154) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_im * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 4.4e+154:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	else:
		tmp = (x_46_im + x_46_im) - (x_46_im * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 4.4e+154)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	else
		tmp = Float64(Float64(x_46_im + x_46_im) - Float64(x_46_im * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 4.4e+154)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	else
		tmp = (x_46_im + x_46_im) - (x_46_im * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 4.4e+154], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] - N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 4.4000000000000002e154

   1. Initial program 84.0%

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

      1. add-cube-cbrt 83.6%

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

      2. pow3 83.7%

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

      3. *-commutative 83.7%

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

      4. difference-of-squares 84.1%

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

      5. associate-*r* 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. unpow3 91.9%

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

      2. add-cube-cbrt 92.2%

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

      3. *-commutative 92.2%

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

      4. +-commutative 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. distribute-lft-out 92.2%

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

   7. Applied egg-rr 92.2%

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

   8. Taylor expanded in x.re around inf 51.2%

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

      1. distribute-rgt1-in 51.2%

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

      2. metadata-eval 51.2%

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

      3. associate-*r* 51.2%

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

      4. unpow2 51.2%

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

      5. associate-*r* 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-*r* 59.0%

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

      8. *-commutative 59.0%

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

   10. Simplified 59.0%

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

   if 4.4000000000000002e154 < x.im

   1. Initial program 46.7%

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

      1. +-commutative 46.7%

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

      2. *-commutative 46.7%

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

      3. fma-def 50.0%

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

      4. *-commutative 50.0%

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

      5. distribute-rgt-out 50.0%

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

      6. *-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. fma-udef 46.7%

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

      2. distribute-lft-in 46.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 56.7%

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

      9. distribute-lft-in 56.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 73.3%

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

      16. difference-of-squares 100.0%

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

      17. associate-*r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 47.2%

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

   7. Taylor expanded in x.re around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. unpow2 53.3%

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

      3. associate-*r* 43.9%

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

      4. *-commutative 43.9%

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

      5. distribute-rgt-neg-in 43.9%

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

      6. *-commutative 43.9%

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

      7. distribute-rgt-neg-in 43.9%

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

   9. Simplified 43.9%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 4.4 \cdot 10^{+154}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.im \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 6: 59.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im 1.5e+150)
   (* (* x.re x.im) (* x.re 3.0))
   (- (+ x.im x.im) (* x.re (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= 1.5e+150) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= 1.5d+150) then
        tmp = (x_46re * x_46im) * (x_46re * 3.0d0)
    else
        tmp = (x_46im + x_46im) - (x_46re * (x_46im * x_46im))
    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 <= 1.5e+150) {
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	} else {
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= 1.5e+150:
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0)
	else:
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= 1.5e+150)
		tmp = Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0));
	else
		tmp = Float64(Float64(x_46_im + x_46_im) - Float64(x_46_re * Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= 1.5e+150)
		tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
	else
		tmp = (x_46_im + x_46_im) - (x_46_re * (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, 1.5e+150], N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] - N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.50000000000000006e150

   1. Initial program 84.0%

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

      1. add-cube-cbrt 83.6%

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

      2. pow3 83.7%

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

      3. *-commutative 83.7%

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

      4. difference-of-squares 84.1%

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

      5. associate-*r* 91.9%

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

   3. Applied egg-rr 91.9%

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

      1. unpow3 91.9%

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

      2. add-cube-cbrt 92.2%

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

      3. *-commutative 92.2%

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

      4. +-commutative 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. *-commutative 92.2%

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

      2. distribute-lft-out 92.2%

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

   7. Applied egg-rr 92.2%

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

   8. Taylor expanded in x.re around inf 51.2%

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

      1. distribute-rgt1-in 51.2%

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

      2. metadata-eval 51.2%

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

      3. associate-*r* 51.2%

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

      4. unpow2 51.2%

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

      5. associate-*r* 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-*r* 59.0%

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

      8. *-commutative 59.0%

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

   10. Simplified 59.0%

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

   if 1.50000000000000006e150 < x.im

   1. Initial program 46.7%

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

      1. +-commutative 46.7%

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

      2. *-commutative 46.7%

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

      3. fma-def 50.0%

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

      4. *-commutative 50.0%

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

      5. distribute-rgt-out 50.0%

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

      6. *-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. fma-udef 46.7%

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

      2. distribute-lft-in 46.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 56.7%

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

      9. distribute-lft-in 56.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 73.3%

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

      16. difference-of-squares 100.0%

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

      17. associate-*r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 47.2%

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

   7. Taylor expanded in x.re around 0 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*r* 53.3%

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

      3. mul-1-neg 53.3%

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

   9. Simplified 53.3%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.5 \cdot 10^{+150}:\\ \;\;\;\;\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) - x.re \cdot \left(x.im \cdot x.im\right)\\ \end{array} \]

Alternative 7: 35.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im) :precision binary64 (* 2.0 (* x.re (* x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	return 2.0 * (x_46_re * (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 = 2.0d0 * (x_46re * (x_46re * x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. add-cube-cbrt 79.3%

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

   2. pow3 79.3%

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

   3. *-commutative 79.3%

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

   4. difference-of-squares 80.9%

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

   5. associate-*r* 87.8%

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

3. Applied egg-rr 87.8%

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

4. Taylor expanded in x.im around 0 31.9%

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

   1. *-commutative 31.9%

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

   2. unpow2 31.9%

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

   3. associate-*r* 32.9%

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

6. Simplified 32.9%

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

7. Final simplification 32.9%

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

Alternative 8: 49.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* (* x.re x.re) 3.0)))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * ((x_46_re * x_46_re) * 3.0);
}

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_46re * x_46re) * 3.0d0)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * ((x_46_re * x_46_re) * 3.0);
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * ((x_46_re * x_46_re) * 3.0)

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in x.re around inf 46.4%

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

4. Simplified 46.4%

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

5. Taylor expanded in x.im around 0 46.4%

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

   1. *-commutative 46.4%

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

   2. distribute-lft1-in 46.4%

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

   3. metadata-eval 46.4%

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

   4. unpow2 46.4%

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

7. Simplified 46.4%

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

8. Final simplification 46.4%

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

Alternative 9: 49.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x.im \cdot \left(x.re \cdot \left(x.re \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* x.im (* x.re (* x.re 3.0))))

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re * 3.0));
}

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_46re * (x_46re * 3.0d0))
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return x_46_im * (x_46_re * (x_46_re * 3.0));
}

Python

def code(x_46_re, x_46_im):
	return x_46_im * (x_46_re * (x_46_re * 3.0))

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im * Float64(x_46_re * Float64(x_46_re * 3.0)))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = x_46_im * (x_46_re * (x_46_re * 3.0));
end

Wolfram

code[x$46$re_, x$46$im_] := N[(x$46$im * N[(x$46$re * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. add-cube-cbrt 79.3%

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

   2. pow3 79.3%

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

   3. *-commutative 79.3%

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

   4. difference-of-squares 80.9%

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

   5. associate-*r* 87.8%

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

3. Applied egg-rr 87.8%

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

   1. unpow3 87.7%

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

   2. add-cube-cbrt 88.1%

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

   3. *-commutative 88.1%

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

   4. +-commutative 88.1%

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

5. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. distribute-lft-out 88.1%

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

7. Applied egg-rr 88.1%

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

8. Taylor expanded in x.re around inf 46.4%

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

   1. distribute-rgt1-in 46.4%

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

   2. metadata-eval 46.4%

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

   3. associate-*r* 46.4%

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

   4. unpow2 46.4%

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

   5. associate-*r* 46.4%

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

   6. *-commutative 46.4%

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

   7. *-commutative 46.4%

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

10. Simplified 46.4%

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

11. Final simplification 46.4%

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

Alternative 10: 55.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(x.re \cdot x.im\right) \cdot \left(x.re \cdot 3\right) \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 (* (* x.re x.im) (* x.re 3.0)))

C

double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_im) * (x_46_re * 3.0);
}

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) * (x_46re * 3.0d0)
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return (x_46_re * x_46_im) * (x_46_re * 3.0);
}

Python

def code(x_46_re, x_46_im):
	return (x_46_re * x_46_im) * (x_46_re * 3.0)

Julia

function code(x_46_re, x_46_im)
	return Float64(Float64(x_46_re * x_46_im) * Float64(x_46_re * 3.0))
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = (x_46_re * x_46_im) * (x_46_re * 3.0);
end

Wolfram

code[x$46$re_, x$46$im_] := N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. add-cube-cbrt 79.3%

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

   2. pow3 79.3%

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

   3. *-commutative 79.3%

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

   4. difference-of-squares 80.9%

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

   5. associate-*r* 87.8%

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

3. Applied egg-rr 87.8%

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

   1. unpow3 87.7%

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

   2. add-cube-cbrt 88.1%

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

   3. *-commutative 88.1%

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

   4. +-commutative 88.1%

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

5. Applied egg-rr 88.1%

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

   1. *-commutative 88.1%

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

   2. distribute-lft-out 88.1%

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

7. Applied egg-rr 88.1%

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

8. Taylor expanded in x.re around inf 46.4%

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

   1. distribute-rgt1-in 46.4%

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

   2. metadata-eval 46.4%

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

   3. associate-*r* 46.4%

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

   4. unpow2 46.4%

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

   5. associate-*r* 46.4%

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

   6. *-commutative 46.4%

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

   7. associate-*r* 53.3%

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

   8. *-commutative 53.3%

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

10. Simplified 53.3%

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

11. Final simplification 53.3%

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

Alternative 11: 34.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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
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;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. *-commutative 79.6%

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

   2. *-commutative 79.6%

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

   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-+ 62.0%

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

   9. distribute-lft-in 62.0%

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

   10. flip-+ 0.0%

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

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. flip-+ 49.4%

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

3. Applied egg-rr 49.4%

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

4. Taylor expanded in x.re around inf 31.5%

   \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]
5. Step-by-step derivation

   1. *-commutative 31.5%

      \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]

   2. unpow2 31.5%

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

6. Simplified 31.5%

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

7. Final simplification 31.5%

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

Alternative 12: 34.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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)
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);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. *-commutative 79.6%

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

   2. *-commutative 79.6%

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

   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-+ 62.0%

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

   9. distribute-lft-in 62.0%

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

   10. flip-+ 0.0%

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

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. flip-+ 49.4%

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

3. Applied egg-rr 49.4%

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

4. Taylor expanded in x.re around inf 31.5%

   \[\leadsto \color{blue}{{x.re}^{2} \cdot x.im} \]
5. Step-by-step derivation

   1. *-commutative 31.5%

      \[\leadsto \color{blue}{x.im \cdot {x.re}^{2}} \]

   2. unpow2 31.5%

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

   3. associate-*r* 32.3%

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

6. Simplified 32.3%

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

7. Final simplification 32.3%

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

Alternative 13: 3.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.im + x.im \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return x_46_im + x_46_im

Julia

function code(x_46_re, x_46_im)
	return Float64(x_46_im + x_46_im)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. add-cube-cbrt 79.3%

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

   2. pow3 79.3%

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

   3. *-commutative 79.3%

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

   4. difference-of-squares 80.9%

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

   5. associate-*r* 87.8%

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

3. Applied egg-rr 87.8%

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

4. Taylor expanded in x.im around 0 31.9%

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

   1. *-commutative 31.9%

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

   2. unpow2 31.9%

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

   3. associate-*r* 32.9%

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

6. Simplified 32.9%

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

   1. associate-*r* 32.9%

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

   2. count-2 32.9%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

      \[\leadsto \frac{\color{blue}{0}}{x.im \cdot x.re - x.im \cdot x.re} \cdot x.re \]

   5. +-inverses 0.0%

      \[\leadsto \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.im \cdot x.re - x.im \cdot x.re} \cdot x.re \]

   6. +-inverses 0.0%

      \[\leadsto \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \cdot x.re \]

   7. +-inverses 0.0%

      \[\leadsto \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \cdot x.re \]

   8. flip-+ 20.2%

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

   9. count-2 20.2%

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

   10. associate-*r* 20.2%

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

   11. count-2 20.2%

       \[\leadsto \color{blue}{x.im \cdot x.re + x.im \cdot x.re} \]

   12. flip-+ 0.0%

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

   13. +-inverses 0.0%

       \[\leadsto \frac{\color{blue}{0}}{x.im \cdot x.re - x.im \cdot x.re} \]

   14. +-inverses 0.0%

       \[\leadsto \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.im \cdot x.re - x.im \cdot x.re} \]

   15. +-inverses 0.0%

       \[\leadsto \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \]

   16. +-inverses 0.0%

       \[\leadsto \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \]

   17. flip-+ 3.0%

       \[\leadsto \color{blue}{x.im + x.im} \]

8. Applied egg-rr 3.0%

   \[\leadsto \color{blue}{x.im + x.im} \]

9. Final simplification 3.0%

   \[\leadsto x.im + x.im \]

Alternative 14: 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 79.6%

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

   1. +-commutative 79.6%

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

   2. *-commutative 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-lft-in 78.1%

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

   5. associate-+r+ 78.1%

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

   6. distribute-rgt-neg-out 78.1%

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

   7. unsub-neg 78.1%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 84.9%

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

   10. *-commutative 84.9%

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

   11. count-2 84.9%

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

   12. distribute-lft1-in 84.9%

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

   13. metadata-eval 84.9%

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

   14. *-commutative 84.9%

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

   15. *-commutative 84.9%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

   1. associate-*r* 85.0%

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

   2. associate-*l* 85.0%

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

   3. flip-- 26.1%

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

   4. div-inv 25.4%

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

   5. swap-sqr 25.4%

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

   6. pow2 25.4%

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

   7. metadata-eval 25.4%

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

   8. pow-prod-up 25.4%

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

   9. metadata-eval 25.4%

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

   10. associate-*l* 25.3%

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

   11. associate-*r* 25.3%

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

   12. fma-def 25.3%

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

5. Applied egg-rr 25.3%

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

7. Simplified 2.7%

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

8. Final simplification 2.7%

   \[\leadsto -10 \]

Alternative 15: 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 79.6%

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

   1. +-commutative 79.6%

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

   2. *-commutative 79.6%

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

   3. sub-neg 79.6%

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

   4. distribute-lft-in 78.1%

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

   5. associate-+r+ 78.1%

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

   6. distribute-rgt-neg-out 78.1%

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

   7. unsub-neg 78.1%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 84.9%

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

   10. *-commutative 84.9%

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

   11. count-2 84.9%

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

   12. distribute-lft1-in 84.9%

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

   13. metadata-eval 84.9%

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

   14. *-commutative 84.9%

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

   15. *-commutative 84.9%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

   1. sub-neg 85.0%

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

   2. associate-*r* 85.0%

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

   3. associate-*l* 85.0%

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

   4. flip3-+ 18.9%

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

   5. associate-*r* 17.0%

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

   6. associate-*r* 16.9%

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

   7. unpow-prod-down 8.8%

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

   8. pow2 8.8%

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

   9. pow-pow 8.8%

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

   10. metadata-eval 8.8%

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

5. Applied egg-rr 8.8%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto 0.1 \]

Developer target: 91.5% 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 2023224 
(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)))