math.cube on complex, imaginary part

Percentage Accurate: 82.7% → 99.7%

Time: 7.7s

Alternatives: 12

Speedup: 1.3×

• 44.4% 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: 82.7% 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: 99.7% 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 95.8%

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

      1. add-cube-cbrt 95.3%

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

      2. pow3 95.3%

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

      3. *-commutative 95.3%

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

      4. difference-of-squares 95.3%

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

      5. associate-*r* 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow3 99.2%

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

      2. add-cube-cbrt 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   7. Applied egg-rr 99.8%

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

   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 12.0%

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

      5. associate-*r* 12.0%

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

   3. Applied egg-rr 12.0%

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

      1. unpow3 12.0%

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

      2. add-cube-cbrt 12.0%

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

      3. *-commutative 12.0%

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

      4. +-commutative 12.0%

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

   5. Applied egg-rr 12.0%

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

      1. *-commutative 12.0%

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

      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: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{-104} \lor \neg \left(x.im \leq 6.2 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.5e-104) (not (<= x.im 6.2e-104)))
   (+ (* (- x.re x.im) (* x.im (+ x.re x.im))) (* x.re 0.0))
   (+ (* x.re (* x.re (+ x.im x.im))) (* (* x.re x.im) (- x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.5e-104) || !(x_46_im <= 6.2e-104)) {
		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_re * (x_46_im + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re - x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.5d-104)) .or. (.not. (x_46im <= 6.2d-104))) then
        tmp = ((x_46re - x_46im) * (x_46im * (x_46re + x_46im))) + (x_46re * 0.0d0)
    else
        tmp = (x_46re * (x_46re * (x_46im + x_46im))) + ((x_46re * x_46im) * (x_46re - x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.5e-104) || !(x_46_im <= 6.2e-104)) {
		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_re * (x_46_im + x_46_im))) + ((x_46_re * 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 <= -2.5e-104) or not (x_46_im <= 6.2e-104):
		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_re * (x_46_im + x_46_im))) + ((x_46_re * 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 <= -2.5e-104) || !(x_46_im <= 6.2e-104))
		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 * Float64(x_46_re * Float64(x_46_im + x_46_im))) + Float64(Float64(x_46_re * 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 <= -2.5e-104) || ~((x_46_im <= 6.2e-104)))
		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_re * (x_46_im + x_46_im))) + ((x_46_re * x_46_im) * (x_46_re - x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.5e-104], N[Not[LessEqual[x$46$im, 6.2e-104]], $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 * N[(x$46$re * N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.49999999999999989e-104 or 6.19999999999999951e-104 < x.im

   1. Initial program 83.5%

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

      1. add-cube-cbrt 82.8%

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

      2. pow3 82.9%

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

      3. *-commutative 82.9%

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

      4. difference-of-squares 84.8%

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

      5. associate-*r* 85.4%

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

   3. Applied egg-rr 85.4%

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

      1. unpow3 85.3%

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

      2. add-cube-cbrt 86.0%

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

      3. *-commutative 86.0%

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

      4. +-commutative 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      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 91.1%

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

   9. Simplified 91.1%

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

   if -2.49999999999999989e-104 < x.im < 6.19999999999999951e-104

   1. Initial program 91.3%

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

      1. add-cube-cbrt 91.0%

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

      2. pow3 91.0%

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

      3. *-commutative 91.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-*r* 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 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in x.im around 0 99.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.5 \cdot 10^{-104} \lor \neg \left(x.im \leq 6.2 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im + x.im\right)\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \end{array} \]

Alternative 3: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -7.4 \cdot 10^{-103} \lor \neg \left(x.im \leq 2.3 \cdot 10^{-103}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -7.4e-103) (not (<= x.im 2.3e-103)))
   (+ (* (- x.re x.im) (* x.im (+ x.re x.im))) (* x.re 0.0))
   (* x.re (* (* x.re x.im) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -7.4e-103) || !(x_46_im <= 2.3e-103)) {
		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_re * x_46_im) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-7.4d-103)) .or. (.not. (x_46im <= 2.3d-103))) then
        tmp = ((x_46re - x_46im) * (x_46im * (x_46re + x_46im))) + (x_46re * 0.0d0)
    else
        tmp = x_46re * ((x_46re * x_46im) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -7.4e-103) || !(x_46_im <= 2.3e-103)) {
		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_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -7.4e-103) or not (x_46_im <= 2.3e-103):
		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_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -7.4e-103) || !(x_46_im <= 2.3e-103))
		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(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -7.4e-103) || ~((x_46_im <= 2.3e-103)))
		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_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -7.4e-103], N[Not[LessEqual[x$46$im, 2.3e-103]], $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[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -7.3999999999999999e-103 or 2.3000000000000001e-103 < x.im

   1. Initial program 83.5%

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

      1. add-cube-cbrt 82.8%

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

      2. pow3 82.9%

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

      3. *-commutative 82.9%

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

      4. difference-of-squares 84.8%

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

      5. associate-*r* 85.4%

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

   3. Applied egg-rr 85.4%

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

      1. unpow3 85.3%

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

      2. add-cube-cbrt 86.0%

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

      3. *-commutative 86.0%

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

      4. +-commutative 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      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 91.1%

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

   9. Simplified 91.1%

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

   if -7.3999999999999999e-103 < x.im < 2.3000000000000001e-103

   1. Initial program 91.3%

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

      1. add-cube-cbrt 91.0%

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

      2. pow3 91.0%

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

      3. *-commutative 91.0%

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

      4. difference-of-squares 91.0%

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

      5. associate-*r* 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 99.8%

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

      3. *-commutative 99.8%

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

      4. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around inf 91.2%

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

      1. distribute-rgt1-in 91.2%

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

      2. metadata-eval 91.2%

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

      3. *-commutative 91.2%

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

      4. associate-*r* 91.2%

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

      5. unpow2 91.2%

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

      6. associate-*r* 99.8%

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

      7. associate-*l* 99.8%

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

      8. *-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.4 \cdot 10^{-103} \lor \neg \left(x.im \leq 2.3 \cdot 10^{-103}\right):\\ \;\;\;\;\left(x.re - x.im\right) \cdot \left(x.im \cdot \left(x.re + x.im\right)\right) + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -15000000000 \lor \neg \left(x.im \leq 4.5 \cdot 10^{+22}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -15000000000.0) (not (<= x.im 4.5e+22)))
   (+ (+ x.im x.im) (* (* x.im x.im) (- x.re x.im)))
   (* x.re (* (* x.re x.im) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -15000000000.0) || !(x_46_im <= 4.5e+22)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-15000000000.0d0)) .or. (.not. (x_46im <= 4.5d+22))) then
        tmp = (x_46im + x_46im) + ((x_46im * x_46im) * (x_46re - x_46im))
    else
        tmp = x_46re * ((x_46re * x_46im) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -15000000000.0) || !(x_46_im <= 4.5e+22)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	} else {
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -15000000000.0) or not (x_46_im <= 4.5e+22):
		tmp = (x_46_im + x_46_im) + ((x_46_im * x_46_im) * (x_46_re - x_46_im))
	else:
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -15000000000.0) || !(x_46_im <= 4.5e+22))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im * x_46_im) * Float64(x_46_re - x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -15000000000.0) || ~((x_46_im <= 4.5e+22)))
		tmp = (x_46_im + x_46_im) + ((x_46_im * x_46_im) * (x_46_re - x_46_im));
	else
		tmp = x_46_re * ((x_46_re * x_46_im) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -15000000000.0], N[Not[LessEqual[x$46$im, 4.5e+22]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.5e10 or 4.4999999999999998e22 < x.im

   1. Initial program 76.2%

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

      1. +-commutative 76.2%

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

      2. *-commutative 76.2%

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

      3. fma-def 80.9%

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

      4. *-commutative 80.9%

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

      5. distribute-rgt-out 80.9%

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

      6. *-commutative 80.9%

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

   3. Simplified 80.9%

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

      1. fma-udef 76.2%

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

      2. distribute-lft-in 76.2%

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

      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-+ 84.3%

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

      9. distribute-lft-in 84.3%

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

      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-+ 88.1%

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

      16. difference-of-squares 97.6%

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

      17. associate-*r* 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in x.im around inf 84.6%

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

      1. unpow2 84.6%

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

   8. Simplified 84.6%

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

   if -1.5e10 < x.im < 4.4999999999999998e22

   1. Initial program 93.7%

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

      1. add-cube-cbrt 93.0%

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

      2. pow3 93.1%

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

      3. *-commutative 93.1%

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

      4. difference-of-squares 93.1%

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

      5. associate-*r* 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. unpow3 99.1%

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

      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. Taylor expanded in x.re around inf 76.6%

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

      1. distribute-rgt1-in 76.6%

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

      2. metadata-eval 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*r* 76.6%

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

      5. unpow2 76.6%

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

      6. associate-*r* 82.7%

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

      7. associate-*l* 82.8%

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

      8. *-commutative 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -15000000000 \lor \neg \left(x.im \leq 4.5 \cdot 10^{+22}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot x.im\right) \cdot \left(x.re - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 5: 36.0% 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 86.5%

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

   1. add-cube-cbrt 86.0%

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

   2. pow3 86.0%

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

   3. *-commutative 86.0%

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

   4. difference-of-squares 87.2%

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

   5. associate-*r* 90.7%

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

3. Applied egg-rr 90.7%

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

4. Taylor expanded in x.im around 0 36.4%

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

   1. *-commutative 36.4%

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

   2. unpow2 36.4%

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

   3. associate-*r* 36.9%

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

6. Simplified 36.9%

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

7. Final simplification 36.9%

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

Alternative 6: 50.5% 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 86.5%

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

2. Taylor expanded in x.re around inf 53.0%

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

4. Simplified 53.0%

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

5. Taylor expanded in x.im around 0 52.9%

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

   1. *-commutative 52.9%

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

   2. distribute-lft1-in 52.9%

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

   3. metadata-eval 52.9%

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

   4. unpow2 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Alternative 7: 56.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * ((x_46_re * x_46_im) * 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_46re * x_46im) * 3.0d0)
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) * 3.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. add-cube-cbrt 86.0%

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

   2. pow3 86.0%

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

   3. *-commutative 86.0%

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

   4. difference-of-squares 87.2%

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

   5. associate-*r* 90.7%

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

3. Applied egg-rr 90.7%

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

   1. unpow3 90.7%

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

   2. add-cube-cbrt 91.2%

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

   3. *-commutative 91.2%

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

   4. +-commutative 91.2%

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

5. Applied egg-rr 91.2%

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

6. Taylor expanded in x.re around inf 52.9%

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

   1. distribute-rgt1-in 52.9%

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

   2. metadata-eval 52.9%

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

   3. *-commutative 52.9%

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

   4. associate-*r* 52.9%

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

   5. unpow2 52.9%

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

   6. associate-*r* 56.5%

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

   7. associate-*l* 56.5%

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

   8. *-commutative 56.5%

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

8. Simplified 56.5%

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

9. Final simplification 56.5%

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

Alternative 8: 34.7% 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 86.5%

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

   1. *-commutative 86.5%

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

   2. *-commutative 86.5%

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

   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-+ 66.7%

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

   9. distribute-lft-in 66.7%

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

   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-+ 50.0%

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

3. Applied egg-rr 50.0%

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

4. Taylor expanded in x.re around inf 35.9%

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

   1. *-commutative 35.9%

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

   2. unpow2 35.9%

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

6. Simplified 35.9%

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

7. Final simplification 35.9%

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

Alternative 9: 35.4% 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 86.5%

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

   1. *-commutative 86.5%

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

   2. *-commutative 86.5%

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

   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-+ 66.7%

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

   9. distribute-lft-in 66.7%

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

   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-+ 50.0%

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

3. Applied egg-rr 50.0%

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

4. Taylor expanded in x.re around inf 35.9%

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

   1. *-commutative 35.9%

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

   2. unpow2 35.9%

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

   3. associate-*r* 36.3%

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

6. Simplified 36.3%

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

7. Final simplification 36.3%

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

Alternative 10: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -10.0

Julia

function code(x_46_re, x_46_im)
	return -10.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. +-commutative 86.5%

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

   2. *-commutative 86.5%

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

   3. sub-neg 86.5%

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

   4. distribute-lft-in 84.5%

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

   5. associate-+r+ 84.5%

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

   6. distribute-rgt-neg-out 84.5%

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

   7. unsub-neg 84.5%

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

   8. associate-*r* 88.1%

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

   9. distribute-rgt-out 88.1%

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

   10. *-commutative 88.1%

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

   11. count-2 88.1%

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

   12. distribute-lft1-in 88.1%

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

   13. metadata-eval 88.1%

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

   14. *-commutative 88.1%

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

   15. *-commutative 88.1%

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

   16. associate-*r* 88.1%

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

   17. cube-unmult 88.1%

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

3. Simplified 88.1%

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

   1. associate-*r* 88.2%

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

   2. associate-*l* 88.2%

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

   3. flip-- 23.7%

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

   4. div-inv 23.4%

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

   5. swap-sqr 23.3%

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

   6. pow2 23.3%

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

   7. metadata-eval 23.3%

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

   8. pow-prod-up 23.3%

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

   9. metadata-eval 23.3%

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

   10. associate-*l* 23.2%

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

   11. associate-*r* 23.2%

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

   12. fma-def 23.2%

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

5. Applied egg-rr 23.2%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto -10 \]

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.0d0
end function

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in x.re around 0 61.5%

   \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]

4. Simplified 2.8%

   \[\leadsto \color{blue}{-3} \]

5. Final simplification 2.8%

   \[\leadsto -3 \]

Alternative 12: 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 86.5%

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

   1. +-commutative 86.5%

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

   2. *-commutative 86.5%

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

   3. sub-neg 86.5%

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

   4. distribute-lft-in 84.5%

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

   5. associate-+r+ 84.5%

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

   6. distribute-rgt-neg-out 84.5%

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

   7. unsub-neg 84.5%

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

   8. associate-*r* 88.1%

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

   9. distribute-rgt-out 88.1%

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

   10. *-commutative 88.1%

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

   11. count-2 88.1%

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

   12. distribute-lft1-in 88.1%

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

   13. metadata-eval 88.1%

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

   14. *-commutative 88.1%

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

   15. *-commutative 88.1%

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

   16. associate-*r* 88.1%

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

   17. cube-unmult 88.1%

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

3. Simplified 88.1%

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

   1. sub-neg 88.1%

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

   2. associate-*r* 88.2%

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

   3. associate-*l* 88.2%

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

   4. flip3-+ 16.5%

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

   5. associate-*r* 16.1%

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

   6. associate-*r* 16.1%

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

   7. unpow-prod-down 10.7%

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

   8. pow2 10.7%

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

   9. pow-pow 10.7%

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

   10. metadata-eval 10.7%

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

5. Applied egg-rr 10.7%

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

7. Simplified 2.9%

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

8. Final simplification 2.9%

   \[\leadsto 0.1 \]

Developer target: 91.6% 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 2023199 
(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)))