math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 94.1%

Time: 6.0s

Alternatives: 12

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% 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: 94.1% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (- (* x.re (* x.re (* x.im 3.0))) (pow 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 * ((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 = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	} else {
		tmp = -pow(x_46_im, 3.0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double 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 = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = -Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(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 = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	else:
		tmp = -math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(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(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	else
		tmp = Float64(-(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 * ((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 = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	else
		tmp = -(x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := 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[(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], (-N[Power[x$46$im, 3.0], $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 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. +-commutative 93.7%

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

      2. *-commutative 93.7%

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

      3. sub-neg 93.7%

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

      4. distribute-lft-in 91.5%

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

      5. associate-+r+ 91.5%

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

      6. distribute-rgt-neg-out 91.5%

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

      7. unsub-neg 91.5%

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

      8. associate-*r* 97.6%

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

      9. distribute-rgt-out 97.6%

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

      10. *-commutative 97.6%

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

      11. count-2 97.6%

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

      12. distribute-lft1-in 97.6%

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

      13. metadata-eval 97.6%

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

      14. *-commutative 97.6%

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

      15. *-commutative 97.6%

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

      16. associate-*r* 97.6%

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

      17. cube-unmult 97.7%

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

   3. Simplified 97.7%

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

   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. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      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-+ 42.3%

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

      9. distribute-lft-in 42.3%

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

   3. Applied egg-rr 42.3%

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

      1. *-commutative 42.3%

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

      2. distribute-rgt-out 42.3%

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

      3. distribute-lft-out 53.8%

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

   5. Applied egg-rr 53.8%

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

   6. Taylor expanded in x.im around inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. mul-1-neg 73.1%

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

   8. Simplified 73.1%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

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

Alternative 2: 94.1% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<=
      (+
       (* x.im (- (* x.re x.re) (* x.im x.im)))
       (* x.re (+ (* x.re x.im) (* x.re x.im))))
      INFINITY)
   (- (* x.re (* (* x.re x.im) 3.0)) (pow 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 * ((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 = (x_46_re * ((x_46_re * x_46_im) * 3.0)) - pow(x_46_im, 3.0);
	} else {
		tmp = -pow(x_46_im, 3.0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double 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 = (x_46_re * ((x_46_re * x_46_im) * 3.0)) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = -Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(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 = (x_46_re * ((x_46_re * x_46_im) * 3.0)) - math.pow(x_46_im, 3.0)
	else:
		tmp = -math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(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(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) * 3.0)) - (x_46_im ^ 3.0));
	else
		tmp = Float64(-(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 * ((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 = (x_46_re * ((x_46_re * x_46_im) * 3.0)) - (x_46_im ^ 3.0);
	else
		tmp = -(x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := 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[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], (-N[Power[x$46$im, 3.0], $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 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. +-commutative 93.7%

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

      2. *-commutative 93.7%

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

      3. sub-neg 93.7%

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

      4. distribute-lft-in 91.5%

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

      5. associate-+r+ 91.5%

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

      6. distribute-rgt-neg-out 91.5%

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

      7. unsub-neg 91.5%

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

      8. associate-*r* 97.6%

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

      9. distribute-rgt-out 97.6%

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

      10. *-commutative 97.6%

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

      11. count-2 97.6%

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

      12. distribute-lft1-in 97.6%

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

      13. metadata-eval 97.6%

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

      14. *-commutative 97.6%

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

      15. *-commutative 97.6%

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

      16. associate-*r* 97.6%

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

      17. cube-unmult 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x.re around 0 97.7%

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

   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. *-commutative 0.0%

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

      2. *-commutative 0.0%

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

      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-+ 42.3%

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

      9. distribute-lft-in 42.3%

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

   3. Applied egg-rr 42.3%

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

      1. *-commutative 42.3%

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

      2. distribute-rgt-out 42.3%

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

      3. distribute-lft-out 53.8%

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

   5. Applied egg-rr 53.8%

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

   6. Taylor expanded in x.im around inf 73.1%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3}} \]
   7. Step-by-step derivation

      1. mul-1-neg 73.1%

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

   8. Simplified 73.1%

      \[\leadsto \color{blue}{-{x.im}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

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

Alternative 3: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 7 \cdot 10^{+93}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 7e+93)
   (+
    (* x.im (- (* x.re x.re) (* x.im x.im)))
    (* x.re (+ (* x.re 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_re <= 7e+93) {
		tmp = (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)));
	} 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_46re <= 7d+93) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46re * 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_re <= 7e+93) {
		tmp = (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)));
	} 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_re <= 7e+93:
		tmp = (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)))
	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_re <= 7e+93)
		tmp = 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))));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(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_re <= 7e+93)
		tmp = (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)));
	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[LessEqual[x$46$re, 7e+93], 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], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.99999999999999996e93

   1. Initial program 88.6%

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

   if 6.99999999999999996e93 < x.re

   1. Initial program 57.9%

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

   2. Taylor expanded in x.im around 0 65.9%

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

      1. distribute-lft1-in 65.9%

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

      2. metadata-eval 65.9%

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

      3. *-commutative 65.9%

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

      4. pow2 65.9%

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

   4. Applied egg-rr 65.9%

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

      1. add-log-exp 43.0%

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

      2. *-un-lft-identity 43.0%

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

      3. log-prod 43.0%

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

      4. metadata-eval 43.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im}\right) \]

      5. add-log-exp 65.9%

         \[\leadsto 0 + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im} \]

      6. associate-*l* 66.1%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. associate-*l* 80.9%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      8. *-commutative 80.9%

         \[\leadsto 0 + x.re \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot 3\right)}\right) \]

   6. Applied egg-rr 80.9%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 7 \cdot 10^{+93}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.9 \cdot 10^{-56} \lor \neg \left(x.im \leq 9.5 \cdot 10^{-68}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.9e-56) (not (<= x.im 9.5e-68)))
   (* x.im (+ (- (* x.re x.re) (* x.im x.im)) (+ x.re x.re)))
   (* 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 <= -1.9e-56) || !(x_46_im <= 9.5e-68)) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} 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 <= (-1.9d-56)) .or. (.not. (x_46im <= 9.5d-68))) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + (x_46re + x_46re))
    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 <= -1.9e-56) || !(x_46_im <= 9.5e-68)) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	} 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 <= -1.9e-56) or not (x_46_im <= 9.5e-68):
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re))
	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 <= -1.9e-56) || !(x_46_im <= 9.5e-68))
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + Float64(x_46_re + x_46_re)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(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 <= -1.9e-56) || ~((x_46_im <= 9.5e-68)))
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + (x_46_re + x_46_re));
	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, -1.9e-56], N[Not[LessEqual[x$46$im, 9.5e-68]], $MachinePrecision]], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.9000000000000001e-56 or 9.4999999999999997e-68 < x.im

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. *-commutative 83.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 81.1%

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

      9. distribute-lft-in 81.1%

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

   3. Applied egg-rr 81.1%

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

      1. *-commutative 81.1%

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

      2. distribute-rgt-out 81.1%

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

      3. distribute-lft-out 84.8%

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

   5. Applied egg-rr 84.8%

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

   if -1.9000000000000001e-56 < x.im < 9.4999999999999997e-68

   1. Initial program 85.0%

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

   2. Taylor expanded in x.im around 0 79.7%

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

      1. distribute-lft1-in 79.7%

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

      2. metadata-eval 79.7%

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

      3. *-commutative 79.7%

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

      4. pow2 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. add-log-exp 42.7%

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

      2. *-un-lft-identity 42.7%

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

      3. log-prod 42.7%

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

      4. metadata-eval 42.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im}\right) \]

      5. add-log-exp 79.7%

         \[\leadsto 0 + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im} \]

      6. associate-*l* 79.7%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. associate-*l* 94.4%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      8. *-commutative 94.4%

         \[\leadsto 0 + x.re \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot 3\right)}\right) \]

   6. Applied egg-rr 94.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.9 \cdot 10^{-56} \lor \neg \left(x.im \leq 9.5 \cdot 10^{-68}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 5: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.3 \cdot 10^{+186} \lor \neg \left(x.im \leq 10^{+152}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -4.3e+186) (not (<= x.im 1e+152)))
   (* x.re (* x.re (- x.im)))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4.3e+186) || !(x_46_im <= 1e+152)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-4.3d+186)) .or. (.not. (x_46im <= 1d+152))) then
        tmp = x_46re * (x_46re * -x_46im)
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4.3e+186) || !(x_46_im <= 1e+152)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -4.3e+186) or not (x_46_im <= 1e+152):
		tmp = x_46_re * (x_46_re * -x_46_im)
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -4.3e+186) || !(x_46_im <= 1e+152))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -4.3e+186) || ~((x_46_im <= 1e+152)))
		tmp = x_46_re * (x_46_re * -x_46_im);
	else
		tmp = 3.0 * ((x_46_re * 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, -4.3e+186], N[Not[LessEqual[x$46$im, 1e+152]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.3e186 or 1e152 < x.im

   1. Initial program 65.0%

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

   2. Taylor expanded in x.im around 0 7.2%

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

   3. Taylor expanded in x.re around 0 7.2%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in x.re around 0 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

      3. unpow2 30.8%

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

      4. associate-*l* 30.9%

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

   8. Simplified 30.9%

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

   if -4.3e186 < x.im < 1e152

   1. Initial program 90.1%

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

   2. Taylor expanded in x.re around inf 58.6%

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

   3. Taylor expanded in x.im around 0 58.5%

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

      1. unpow2 58.5%

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

   5. Simplified 58.5%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.3 \cdot 10^{+186} \lor \neg \left(x.im \leq 10^{+152}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 6: 60.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{+186} \lor \neg \left(x.im \leq 2.3 \cdot 10^{+147}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -7.8e+186) (not (<= x.im 2.3e+147)))
   (* x.re (* 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 <= -7.8e+186) || !(x_46_im <= 2.3e+147)) {
		tmp = x_46_re * (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 <= (-7.8d+186)) .or. (.not. (x_46im <= 2.3d+147))) then
        tmp = x_46re * (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 <= -7.8e+186) || !(x_46_im <= 2.3e+147)) {
		tmp = x_46_re * (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 <= -7.8e+186) or not (x_46_im <= 2.3e+147):
		tmp = x_46_re * (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 <= -7.8e+186) || !(x_46_im <= 2.3e+147))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-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 <= -7.8e+186) || ~((x_46_im <= 2.3e+147)))
		tmp = x_46_re * (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, -7.8e+186], N[Not[LessEqual[x$46$im, 2.3e+147]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $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 < -7.8000000000000002e186 or 2.2999999999999999e147 < x.im

   1. Initial program 65.0%

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

   2. Taylor expanded in x.im around 0 7.2%

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

   3. Taylor expanded in x.re around 0 7.2%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in x.re around 0 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

      3. unpow2 30.8%

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

      4. associate-*l* 30.9%

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

   8. Simplified 30.9%

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

   if -7.8000000000000002e186 < x.im < 2.2999999999999999e147

   1. Initial program 90.1%

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

   2. Taylor expanded in x.re around inf 58.6%

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

   3. Taylor expanded in x.im around 0 58.5%

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

      1. unpow2 58.5%

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

      2. associate-*l* 65.7%

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

      3. associate-*r* 65.6%

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

      4. *-commutative 65.6%

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

      5. associate-*l* 65.7%

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

   5. Simplified 65.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.8 \cdot 10^{+186} \lor \neg \left(x.im \leq 2.3 \cdot 10^{+147}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.re \cdot x.im\right) \cdot 3\right)\\ \end{array} \]

Alternative 7: 60.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.2 \cdot 10^{+187} \lor \neg \left(x.im \leq 10^{+152}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.2e+187) (not (<= x.im 1e+152)))
   (* x.re (* 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 <= -3.2e+187) || !(x_46_im <= 1e+152)) {
		tmp = x_46_re * (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 <= (-3.2d+187)) .or. (.not. (x_46im <= 1d+152))) then
        tmp = x_46re * (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 <= -3.2e+187) || !(x_46_im <= 1e+152)) {
		tmp = x_46_re * (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 <= -3.2e+187) or not (x_46_im <= 1e+152):
		tmp = x_46_re * (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 <= -3.2e+187) || !(x_46_im <= 1e+152))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(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 <= -3.2e+187) || ~((x_46_im <= 1e+152)))
		tmp = x_46_re * (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, -3.2e+187], N[Not[LessEqual[x$46$im, 1e+152]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.19999999999999993e187 or 1e152 < x.im

   1. Initial program 65.0%

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

   2. Taylor expanded in x.im around 0 7.2%

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

   3. Taylor expanded in x.re around 0 7.2%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in x.re around 0 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

      3. unpow2 30.8%

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

      4. associate-*l* 30.9%

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

   8. Simplified 30.9%

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

   if -3.19999999999999993e187 < x.im < 1e152

   1. Initial program 90.1%

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

   2. Taylor expanded in x.im around 0 58.5%

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

      1. distribute-lft1-in 58.5%

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

      2. metadata-eval 58.5%

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

      3. *-commutative 58.5%

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

      4. pow2 58.5%

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

   4. Applied egg-rr 58.5%

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

      1. add-log-exp 34.9%

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

      2. *-un-lft-identity 34.9%

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

      3. log-prod 34.9%

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

      4. metadata-eval 34.9%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im}\right) \]

      5. add-log-exp 58.5%

         \[\leadsto 0 + \color{blue}{\left(\left(x.re \cdot x.re\right) \cdot 3\right) \cdot x.im} \]

      6. associate-*l* 58.6%

         \[\leadsto 0 + \color{blue}{\left(x.re \cdot x.re\right) \cdot \left(3 \cdot x.im\right)} \]

      7. associate-*l* 65.7%

         \[\leadsto 0 + \color{blue}{x.re \cdot \left(x.re \cdot \left(3 \cdot x.im\right)\right)} \]

      8. *-commutative 65.7%

         \[\leadsto 0 + x.re \cdot \left(x.re \cdot \color{blue}{\left(x.im \cdot 3\right)}\right) \]

   6. Applied egg-rr 65.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.2 \cdot 10^{+187} \lor \neg \left(x.im \leq 10^{+152}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 8: 40.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.65 \cdot 10^{+186} \lor \neg \left(x.im \leq 9.2 \cdot 10^{+151}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.65e+186) (not (<= x.im 9.2e+151)))
   (* x.re (* x.re (- x.im)))
   (* x.re (* x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.65e+186) || !(x_46_im <= 9.2e+151)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (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 <= (-1.65d+186)) .or. (.not. (x_46im <= 9.2d+151))) then
        tmp = x_46re * (x_46re * -x_46im)
    else
        tmp = x_46re * (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 <= -1.65e+186) || !(x_46_im <= 9.2e+151)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.65e+186) or not (x_46_im <= 9.2e+151):
		tmp = x_46_re * (x_46_re * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.65e+186) || !(x_46_im <= 9.2e+151))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * 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 <= -1.65e+186) || ~((x_46_im <= 9.2e+151)))
		tmp = x_46_re * (x_46_re * -x_46_im);
	else
		tmp = x_46_re * (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, -1.65e+186], N[Not[LessEqual[x$46$im, 9.2e+151]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.65000000000000012e186 or 9.2000000000000003e151 < x.im

   1. Initial program 65.0%

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

   2. Taylor expanded in x.im around 0 7.2%

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

   3. Taylor expanded in x.re around 0 7.2%

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

   5. Simplified 30.8%

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

   6. Taylor expanded in x.re around 0 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

      3. unpow2 30.8%

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

      4. associate-*l* 30.9%

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

   8. Simplified 30.9%

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

   if -1.65000000000000012e186 < x.im < 9.2000000000000003e151

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. *-commutative 90.1%

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

      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-+ 63.6%

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

      9. distribute-lft-in 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. *-commutative 63.6%

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

      2. distribute-rgt-out 63.6%

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

      3. distribute-lft-out 65.1%

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

   5. Applied egg-rr 65.1%

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

   6. Taylor expanded in x.re around inf 37.7%

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

      1. unpow2 37.7%

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

      2. associate-*l* 38.6%

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

   8. Simplified 38.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.65 \cdot 10^{+186} \lor \neg \left(x.im \leq 9.2 \cdot 10^{+151}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 9: 35.2% 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 84.2%

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

   1. *-commutative 84.2%

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

   2. *-commutative 84.2%

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

   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-+ 67.4%

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

   9. distribute-lft-in 67.4%

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

3. Applied egg-rr 67.4%

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

   1. *-commutative 67.4%

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

   2. distribute-rgt-out 67.4%

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

   3. distribute-lft-out 69.8%

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

5. Applied egg-rr 69.8%

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

6. Taylor expanded in x.re around inf 30.6%

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

   1. unpow2 30.6%

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

   2. associate-*l* 31.2%

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

8. Simplified 31.2%

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

9. Final simplification 31.2%

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

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 84.2%

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

   1. +-commutative 84.2%

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

   2. *-commutative 84.2%

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

   3. sub-neg 84.2%

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

   4. distribute-lft-in 82.2%

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

   5. associate-+r+ 82.2%

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

   6. distribute-rgt-neg-out 82.2%

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

   7. unsub-neg 82.2%

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

   8. associate-*r* 87.6%

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

   9. distribute-rgt-out 87.7%

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

   10. *-commutative 87.7%

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

   11. count-2 87.7%

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

   12. distribute-lft1-in 87.7%

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

   13. metadata-eval 87.7%

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

   14. *-commutative 87.7%

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

   15. *-commutative 87.7%

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

   16. associate-*r* 87.7%

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

   17. cube-unmult 87.8%

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

3. Simplified 87.8%

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

   1. associate-*r* 87.8%

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

   2. associate-*l* 87.7%

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

   3. flip-- 24.4%

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

   4. div-inv 24.2%

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

   5. swap-sqr 24.2%

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

   6. pow2 24.2%

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

   7. metadata-eval 24.2%

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

   8. pow-prod-up 24.1%

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

   9. metadata-eval 24.1%

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

   10. associate-*l* 24.1%

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

   11. associate-*r* 24.1%

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

   12. fma-def 24.1%

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

5. Applied egg-rr 24.1%

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

7. Simplified 2.7%

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

8. Final simplification 2.7%

   \[\leadsto -10 \]

Alternative 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 84.2%

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

2. Taylor expanded in x.re around 0 60.8%

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

4. Simplified 2.7%

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

5. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 12: 15.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 0.0)

C

double code(double x_46_re, double x_46_im) {
	return 0.0;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 0.0d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 0.0;
}

Python

def code(x_46_re, x_46_im):
	return 0.0

Julia

function code(x_46_re, x_46_im)
	return 0.0
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 0.0;
end

Wolfram

code[x$46$re_, x$46$im_] := 0.0

Derivation

1. Initial program 84.2%

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

   1. *-commutative 84.2%

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

   2. *-commutative 84.2%

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

   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-+ 67.4%

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

   9. distribute-lft-in 67.4%

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

3. Applied egg-rr 67.4%

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

   1. *-commutative 67.4%

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

   2. distribute-rgt-out 67.4%

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

   3. distribute-lft-out 69.8%

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

5. Applied egg-rr 69.8%

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

7. Applied egg-rr 14.1%

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

   1. +-inverses 14.1%

      \[\leadsto \color{blue}{0} \]

9. Simplified 14.1%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 14.1%

    \[\leadsto 0 \]

Developer target: 91.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = ((x_46re * x_46im) * (2.0d0 * x_46re)) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023240 
(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)))