math.cube on complex, imaginary part

Percentage Accurate: 82.3% → 99.7%

Time: 5.8s

Alternatives: 11

Speedup: 1.4×

• 43.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8.2 \cdot 10^{+105} \lor \neg \left(x.im \leq 2 \cdot 10^{+102}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -8.2e+105) (not (<= x.im 2e+102)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 2e+102)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-8.2d+105)) .or. (.not. (x_46im <= 2d+102))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 2e+102)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -8.2e+105) or not (x_46_im <= 2e+102):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 2e+102))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -8.2e+105) || ~((x_46_im <= 2e+102)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -8.2e+105], N[Not[LessEqual[x$46$im, 2e+102]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -8.2000000000000005e105 or 1.99999999999999995e102 < x.im

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 78.4%

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

      8. *-commutative 78.4%

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

      9. distribute-lft-in 78.4%

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

   3. Applied egg-rr 78.4%

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

      1. +-commutative 78.4%

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

      2. add-cube-cbrt 78.4%

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

      3. fma-def 78.4%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -8.2000000000000005e105 < x.im < 1.99999999999999995e102

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. *-commutative 95.9%

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

      3. sub-neg 95.9%

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

      4. distribute-lft-in 95.8%

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

      5. associate-+r+ 95.8%

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

      6. distribute-rgt-neg-out 95.8%

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

      7. unsub-neg 95.8%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8.2 \cdot 10^{+105} \lor \neg \left(x.im \leq 2 \cdot 10^{+102}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8.2 \cdot 10^{+105} \lor \neg \left(x.im \leq 4 \cdot 10^{+83}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -8.2e+105) (not (<= x.im 4e+83)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (- (* x.re (* 3.0 (* x.im x.re))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 4e+83)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-8.2d+105)) .or. (.not. (x_46im <= 4d+83))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = (x_46re * (3.0d0 * (x_46im * x_46re))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 4e+83)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -8.2e+105) or not (x_46_im <= 4e+83):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -8.2e+105) || !(x_46_im <= 4e+83))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -8.2e+105) || ~((x_46_im <= 4e+83)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = (x_46_re * (3.0 * (x_46_im * x_46_re))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -8.2e+105], N[Not[LessEqual[x$46$im, 4e+83]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -8.2000000000000005e105 or 4.00000000000000012e83 < x.im

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 78.4%

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

      8. *-commutative 78.4%

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

      9. distribute-lft-in 78.4%

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

   3. Applied egg-rr 78.4%

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

      1. +-commutative 78.4%

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

      2. add-cube-cbrt 78.4%

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

      3. fma-def 78.4%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -8.2000000000000005e105 < x.im < 4.00000000000000012e83

   1. Initial program 95.9%

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

      1. +-commutative 95.9%

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

      2. *-commutative 95.9%

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

      3. sub-neg 95.9%

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

      4. distribute-lft-in 95.8%

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

      5. associate-+r+ 95.8%

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

      6. distribute-rgt-neg-out 95.8%

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

      7. unsub-neg 95.8%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8.2 \cdot 10^{+105} \lor \neg \left(x.im \leq 4 \cdot 10^{+83}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{if}\;t_0 \leq 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.im x.re) (* x.im x.re))))))
   (if (<= t_0 1e+246)
     t_0
     (+ (+ x.im x.im) (* (+ x.im x.re) (* x.im (- x.re x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (t_0 <= 1e+246) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46im * x_46re) + (x_46im * x_46re)))
    if (t_0 <= 1d+246) then
        tmp = t_0
    else
        tmp = (x_46im + 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 t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (t_0 <= 1e+246) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + 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):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	tmp = 0
	if t_0 <= 1e+246:
		tmp = t_0
	else:
		tmp = (x_46_im + 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)
	t_0 = 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_im * x_46_re) + Float64(x_46_im * x_46_re))))
	tmp = 0.0
	if (t_0 <= 1e+246)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im + x_46_re) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	tmp = 0.0;
	if (t_0 <= 1e+246)
		tmp = t_0;
	else
		tmp = (x_46_im + 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_] := Block[{t$95$0 = 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$46$im * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+246], t$95$0, N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$im + x$46$re), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $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)) < 1.00000000000000007e246

   1. Initial program 99.2%

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

   if 1.00000000000000007e246 < (+.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 64.6%

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

      1. +-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. fma-def 65.9%

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

      4. *-commutative 65.9%

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

      5. distribute-rgt-out 65.9%

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

      6. *-commutative 65.9%

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

   3. Simplified 65.9%

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

      1. fma-udef 64.6%

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

      2. distribute-lft-in 64.6%

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

      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-+ 69.7%

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

      9. distribute-lft-in 69.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 76.2%

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

      16. *-commutative 76.2%

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

      17. difference-of-squares 92.8%

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

      18. associate-*l* 93.6%

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

   5. Applied egg-rr 93.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right) \leq 10^{+246}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im + x.re\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.5 \cdot 10^{-66} \lor \neg \left(x.im \leq 3.1 \cdot 10^{-95}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -9.5e-66) (not (<= x.im 3.1e-95)))
   (* x.im (* (- x.re x.im) (+ x.im x.re)))
   (* x.re (* 3.0 (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -9.5e-66) || !(x_46_im <= 3.1e-95)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	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 <= (-9.5d-66)) .or. (.not. (x_46im <= 3.1d-95))) then
        tmp = x_46im * ((x_46re - x_46im) * (x_46im + x_46re))
    else
        tmp = x_46re * (3.0d0 * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -9.5e-66) || !(x_46_im <= 3.1e-95)) {
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	} else {
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -9.5e-66) or not (x_46_im <= 3.1e-95):
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))
	else:
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -9.5e-66) || !(x_46_im <= 3.1e-95))
		tmp = Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re)));
	else
		tmp = Float64(x_46_re * Float64(3.0 * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -9.5e-66) || ~((x_46_im <= 3.1e-95)))
		tmp = x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re));
	else
		tmp = x_46_re * (3.0 * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -9.5e-66], N[Not[LessEqual[x$46$im, 3.1e-95]], $MachinePrecision]], N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(3.0 * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -9.5000000000000004e-66 or 3.09999999999999992e-95 < x.im

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 76.2%

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

      8. *-commutative 76.2%

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

      9. distribute-lft-in 76.2%

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

   3. Applied egg-rr 76.2%

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

      1. +-commutative 76.2%

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

      2. add-cube-cbrt 76.2%

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

      3. fma-def 76.2%

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

   5. Applied egg-rr 0.0%

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

   7. Simplified 83.2%

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

      1. difference-of-squares 91.1%

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

      2. *-commutative 91.1%

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

   9. Applied egg-rr 91.1%

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

   if -9.5000000000000004e-66 < x.im < 3.09999999999999992e-95

   1. Initial program 92.7%

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

      1. +-commutative 92.7%

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

      2. *-commutative 92.7%

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

      3. sub-neg 92.7%

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

      4. distribute-lft-in 92.7%

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

      5. associate-+r+ 92.7%

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

      6. distribute-rgt-neg-out 92.7%

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

      7. unsub-neg 92.7%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.8%

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

      10. *-commutative 99.8%

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

      11. count-2 99.8%

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

      12. distribute-lft1-in 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. *-commutative 99.8%

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

      16. associate-*r* 99.8%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

   5. Taylor expanded in x.re around inf 92.6%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in x.im around 0 92.6%

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

   10. Simplified 92.6%

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

   11. Taylor expanded in x.im around 0 92.6%

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

       1. unpow2 92.6%

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

       2. associate-*r* 99.7%

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

       3. *-commutative 99.7%

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

       4. *-commutative 99.7%

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

       5. associate-*r* 99.8%

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

       6. *-commutative 99.8%

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

       7. *-commutative 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.5 \cdot 10^{-66} \lor \neg \left(x.im \leq 3.1 \cdot 10^{-95}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 50.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_im * (3.0 * (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_46im * (3.0d0 * (x_46re * x_46re))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. sub-neg 88.7%

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

   4. distribute-lft-in 82.4%

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

   5. associate-+r+ 82.4%

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

   6. distribute-rgt-neg-out 82.4%

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

   7. unsub-neg 82.4%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 85.0%

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

   10. *-commutative 85.0%

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

   11. count-2 85.0%

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

   12. distribute-lft1-in 85.0%

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

   13. metadata-eval 85.0%

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

   14. *-commutative 85.0%

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

   15. *-commutative 85.0%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in x.re around 0 85.0%

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

5. Taylor expanded in x.re around inf 56.0%

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

7. Simplified 56.1%

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

8. Taylor expanded in x.im around 0 56.0%

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

10. Simplified 56.0%

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

11. Final simplification 56.0%

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

Alternative 6: 50.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. sub-neg 88.7%

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

   4. distribute-lft-in 82.4%

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

   5. associate-+r+ 82.4%

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

   6. distribute-rgt-neg-out 82.4%

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

   7. unsub-neg 82.4%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 85.0%

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

   10. *-commutative 85.0%

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

   11. count-2 85.0%

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

   12. distribute-lft1-in 85.0%

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

   13. metadata-eval 85.0%

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

   14. *-commutative 85.0%

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

   15. *-commutative 85.0%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in x.re around 0 85.0%

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

5. Taylor expanded in x.re around inf 56.0%

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

7. Simplified 56.1%

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

8. Final simplification 56.1%

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

Alternative 7: 56.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (3.0 * (x_46_im * 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 * (3.0d0 * (x_46im * x_46re))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. sub-neg 88.7%

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

   4. distribute-lft-in 82.4%

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

   5. associate-+r+ 82.4%

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

   6. distribute-rgt-neg-out 82.4%

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

   7. unsub-neg 82.4%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 85.0%

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

   10. *-commutative 85.0%

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

   11. count-2 85.0%

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

   12. distribute-lft1-in 85.0%

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

   13. metadata-eval 85.0%

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

   14. *-commutative 85.0%

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

   15. *-commutative 85.0%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in x.re around 0 85.0%

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

5. Taylor expanded in x.re around inf 56.0%

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

7. Simplified 56.1%

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

8. Taylor expanded in x.im around 0 56.0%

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

10. Simplified 56.0%

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

11. Taylor expanded in x.im around 0 56.0%

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

    1. unpow2 56.0%

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

    2. associate-*r* 58.5%

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

    3. *-commutative 58.5%

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

    4. *-commutative 58.5%

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

    5. associate-*r* 58.6%

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

    6. *-commutative 58.6%

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

    7. *-commutative 58.6%

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

13. Simplified 58.6%

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

14. Final simplification 58.6%

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

Alternative 8: 34.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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_46im * (x_46re * x_46re)
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);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. *-commutative 88.7%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. flip-+ 66.8%

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

   8. *-commutative 66.8%

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

   9. distribute-lft-in 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x.re around inf 39.0%

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

6. Simplified 39.0%

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

7. Final simplification 39.0%

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

Alternative 9: 35.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return x_46_re * (x_46_im * 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_46im * x_46re)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. *-commutative 88.7%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. flip-+ 66.8%

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

   8. *-commutative 66.8%

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

   9. distribute-lft-in 66.8%

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

3. Applied egg-rr 66.8%

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

4. Taylor expanded in x.re around inf 39.0%

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

6. Simplified 39.0%

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

7. Taylor expanded in x.im around 0 39.0%

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

   1. unpow2 39.0%

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

   2. associate-*r* 39.3%

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

9. Simplified 39.3%

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

10. Final simplification 39.3%

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

Alternative 10: 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 88.7%

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

2. Taylor expanded in x.re around 0 56.0%

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

4. Simplified 2.7%

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

5. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 11: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 0.125)

C

double code(double x_46_re, double x_46_im) {
	return 0.125;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.125d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.125;
}

Python

def code(x_46_re, x_46_im):
	return 0.125

Julia

function code(x_46_re, x_46_im)
	return 0.125
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.125;
end

Wolfram

code[x$46$re_, x$46$im_] := 0.125

Derivation

1. Initial program 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. sub-neg 88.7%

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

   4. distribute-lft-in 82.4%

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

   5. associate-+r+ 82.4%

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

   6. distribute-rgt-neg-out 82.4%

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

   7. unsub-neg 82.4%

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

   8. associate-*r* 84.9%

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

   9. distribute-rgt-out 85.0%

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

   10. *-commutative 85.0%

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

   11. count-2 85.0%

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

   12. distribute-lft1-in 85.0%

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

   13. metadata-eval 85.0%

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

   14. *-commutative 85.0%

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

   15. *-commutative 85.0%

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

   16. associate-*r* 85.0%

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

   17. cube-unmult 85.0%

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

3. Simplified 85.0%

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

   1. sub-neg 85.0%

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

   2. flip3-+ 16.2%

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

   3. associate-*r* 16.2%

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

   4. unpow-prod-down 10.0%

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

   5. pow2 10.0%

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

   6. pow-pow 10.0%

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

   7. metadata-eval 10.0%

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

5. Applied egg-rr 9.9%

   \[\leadsto \color{blue}{\frac{{x.re}^{6} \cdot {\left(x.im \cdot 3\right)}^{3} + {\left(-{x.im}^{3}\right)}^{3}}{{x.re}^{4} \cdot \left(\left(x.im \cdot x.im\right) \cdot 9\right) + \left(\left(-{x.im}^{3}\right) \cdot \left(-{x.im}^{3}\right) - \left(\left(x.re \cdot x.re\right) \cdot \left(x.im \cdot 3\right)\right) \cdot \left(-{x.im}^{3}\right)\right)}} \]

7. Simplified 2.9%

   \[\leadsto \color{blue}{0.125} \]

8. Final simplification 2.9%

   \[\leadsto 0.125 \]

Developer target: 91.2% 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 2023189 
(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)))