math.cube on complex, imaginary part

Percentage Accurate: 82.5% → 98.9%

Time: 8.8s

Alternatives: 18

Speedup: 1.3×

• 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: 82.5% 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: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 6 \cdot 10^{+82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -9.2e+129) (not (<= x.im 6e+82)))
   (+ (* x.im (* (- x.re x.im) (+ x.im x.re))) (+ x.im x.im))
   (- (* (* 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 <= -9.2e+129) || !(x_46_im <= 6e+82)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= (-9.2d+129)) .or. (.not. (x_46im <= 6d+82))) then
        tmp = (x_46im * ((x_46re - x_46im) * (x_46im + x_46re))) + (x_46im + x_46im)
    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 <= -9.2e+129) || !(x_46_im <= 6e+82)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= -9.2e+129) or not (x_46_im <= 6e+82):
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im)
	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 <= -9.2e+129) || !(x_46_im <= 6e+82))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))) + Float64(x_46_im + x_46_im));
	else
		tmp = Float64(Float64(Float64(x_46_re * 3.0) * Float64(x_46_im * x_46_re)) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -9.2e+129) || ~((x_46_im <= 6e+82)))
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	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, -9.2e+129], N[Not[LessEqual[x$46$im, 6e+82]], $MachinePrecision]], N[(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$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * 3.0), $MachinePrecision] * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -9.19999999999999961e129 or 5.99999999999999978e82 < x.im

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      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-+ 68.5%

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

      8. *-commutative 68.5%

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

      9. distribute-lft-in 68.5%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -9.19999999999999961e129 < x.im < 5.99999999999999978e82

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. *-commutative 93.8%

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

      3. sub-neg 93.8%

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

      4. distribute-lft-in 93.8%

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

      5. associate-+r+ 93.8%

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

      6. distribute-rgt-neg-out 93.8%

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

      7. unsub-neg 93.8%

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

      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%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. fma-neg 99.9%

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

      4. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. unsub-neg 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 6 \cdot 10^{+82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 4 \cdot 10^{+70}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -9.2e+129) (not (<= x.im 4e+70)))
   (+ (* x.im (* (- x.re x.im) (+ x.im x.re))) (+ x.im x.im))
   (- (* 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 <= -9.2e+129) || !(x_46_im <= 4e+70)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= (-9.2d+129)) .or. (.not. (x_46im <= 4d+70))) then
        tmp = (x_46im * ((x_46re - x_46im) * (x_46im + x_46re))) + (x_46im + x_46im)
    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 <= -9.2e+129) || !(x_46_im <= 4e+70)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= -9.2e+129) or not (x_46_im <= 4e+70):
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im)
	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 <= -9.2e+129) || !(x_46_im <= 4e+70))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))) + Float64(x_46_im + x_46_im));
	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 <= -9.2e+129) || ~((x_46_im <= 4e+70)))
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	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, -9.2e+129], N[Not[LessEqual[x$46$im, 4e+70]], $MachinePrecision]], N[(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$46$im + x$46$im), $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 < -9.19999999999999961e129 or 4.00000000000000029e70 < x.im

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      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-+ 68.5%

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

      8. *-commutative 68.5%

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

      9. distribute-lft-in 68.5%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -9.19999999999999961e129 < x.im < 4.00000000000000029e70

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. *-commutative 93.8%

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

      3. sub-neg 93.8%

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

      4. distribute-lft-in 93.8%

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

      5. associate-+r+ 93.8%

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

      6. distribute-rgt-neg-out 93.8%

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

      7. unsub-neg 93.8%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 4 \cdot 10^{+70}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(3 \cdot \left(x.im \cdot x.re\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 7 \cdot 10^{+82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -9.2e+129) (not (<= x.im 7e+82)))
   (+ (* x.im (* (- x.re x.im) (+ x.im x.re))) (+ x.im x.im))
   (- (* 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 <= -9.2e+129) || !(x_46_im <= 7e+82)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= (-9.2d+129)) .or. (.not. (x_46im <= 7d+82))) then
        tmp = (x_46im * ((x_46re - x_46im) * (x_46im + x_46re))) + (x_46im + x_46im)
    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 <= -9.2e+129) || !(x_46_im <= 7e+82)) {
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	} 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 <= -9.2e+129) or not (x_46_im <= 7e+82):
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im)
	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 <= -9.2e+129) || !(x_46_im <= 7e+82))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))) + Float64(x_46_im + x_46_im));
	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 <= -9.2e+129) || ~((x_46_im <= 7e+82)))
		tmp = (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re))) + (x_46_im + x_46_im);
	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, -9.2e+129], N[Not[LessEqual[x$46$im, 7e+82]], $MachinePrecision]], N[(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$46$im + x$46$im), $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 < -9.19999999999999961e129 or 7.0000000000000001e82 < x.im

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      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-+ 68.5%

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

      8. *-commutative 68.5%

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

      9. distribute-lft-in 68.5%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 82.2%

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

   3. Applied egg-rr 82.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -9.19999999999999961e129 < x.im < 7.0000000000000001e82

   1. Initial program 93.8%

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

      1. +-commutative 93.8%

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

      2. *-commutative 93.8%

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

      3. sub-neg 93.8%

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

      4. distribute-lft-in 93.8%

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

      5. associate-+r+ 93.8%

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

      6. distribute-rgt-neg-out 93.8%

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

      7. unsub-neg 93.8%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.2 \cdot 10^{+129} \lor \neg \left(x.im \leq 7 \cdot 10^{+82}\right):\\ \;\;\;\;x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{if}\;x.re \leq -6.9 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 2.25 \cdot 10^{-48}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.re\right) - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 2.32 \cdot 10^{-14}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.36 \cdot 10^{+103}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* 3.0 (* x.re (* x.im x.re)))))
   (if (<= x.re -6.9e-28)
     t_0
     (if (<= x.re 2.25e-48)
       (* x.im (- (* x.re (- x.re)) (* x.im x.im)))
       (if (<= x.re 2.32e-14)
         (* 3.0 (* x.im (* x.re x.re)))
         (if (<= x.re 1.36e+103)
           (* x.im (+ (- (* x.re x.re) (* x.im x.im)) 2.0))
           t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	double tmp;
	if (x_46_re <= -6.9e-28) {
		tmp = t_0;
	} else if (x_46_re <= 2.25e-48) {
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	} else if (x_46_re <= 2.32e-14) {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_re <= 1.36e+103) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + 2.0);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (x_46re * (x_46im * x_46re))
    if (x_46re <= (-6.9d-28)) then
        tmp = t_0
    else if (x_46re <= 2.25d-48) then
        tmp = x_46im * ((x_46re * -x_46re) - (x_46im * x_46im))
    else if (x_46re <= 2.32d-14) then
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    else if (x_46re <= 1.36d+103) then
        tmp = x_46im * (((x_46re * x_46re) - (x_46im * x_46im)) + 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	double tmp;
	if (x_46_re <= -6.9e-28) {
		tmp = t_0;
	} else if (x_46_re <= 2.25e-48) {
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	} else if (x_46_re <= 2.32e-14) {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_re <= 1.36e+103) {
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re))
	tmp = 0
	if x_46_re <= -6.9e-28:
		tmp = t_0
	elif x_46_re <= 2.25e-48:
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im))
	elif x_46_re <= 2.32e-14:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	elif x_46_re <= 1.36e+103:
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_re)))
	tmp = 0.0
	if (x_46_re <= -6.9e-28)
		tmp = t_0;
	elseif (x_46_re <= 2.25e-48)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * Float64(-x_46_re)) - Float64(x_46_im * x_46_im)));
	elseif (x_46_re <= 2.32e-14)
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	elseif (x_46_re <= 1.36e+103)
		tmp = Float64(x_46_im * Float64(Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)) + 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	tmp = 0.0;
	if (x_46_re <= -6.9e-28)
		tmp = t_0;
	elseif (x_46_re <= 2.25e-48)
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	elseif (x_46_re <= 2.32e-14)
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	elseif (x_46_re <= 1.36e+103)
		tmp = x_46_im * (((x_46_re * x_46_re) - (x_46_im * x_46_im)) + 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -6.9e-28], t$95$0, If[LessEqual[x$46$re, 2.25e-48], N[(x$46$im * N[(N[(x$46$re * (-x$46$re)), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.32e-14], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.36e+103], N[(x$46$im * N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -6.90000000000000001e-28 or 1.36e103 < x.re

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. *-commutative 71.3%

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

      3. sub-neg 71.3%

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

      4. distribute-lft-in 65.6%

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

      5. associate-+r+ 65.6%

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

      6. distribute-rgt-neg-out 65.6%

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

      7. unsub-neg 65.6%

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

      8. associate-*r* 76.0%

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

      9. distribute-rgt-out 76.0%

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

      10. *-commutative 76.0%

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

      11. count-2 76.0%

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

      12. distribute-lft1-in 76.0%

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

      13. metadata-eval 76.0%

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

      14. *-commutative 76.0%

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

      15. *-commutative 76.0%

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

      16. associate-*r* 76.0%

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

      17. cube-unmult 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x.re around 0 76.0%

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

   5. Taylor expanded in x.re around inf 76.0%

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

      1. unpow2 76.0%

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

      2. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   9. Applied egg-rr 86.5%

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

   if -6.90000000000000001e-28 < x.re < 2.24999999999999994e-48

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. associate-+r+ 99.9%

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

      6. distribute-rgt-neg-out 99.9%

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

      7. unsub-neg 99.9%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 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* 99.9%

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

      17. cube-unmult 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x.re around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. unsub-neg 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow3 99.8%

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

      3. distribute-rgt-out-- 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

   12. Simplified 97.9%

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

   if 2.24999999999999994e-48 < x.re < 2.32000000000000015e-14

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. associate-+r+ 99.4%

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

      6. distribute-rgt-neg-out 99.4%

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

      7. unsub-neg 99.4%

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

      8. associate-*r* 99.0%

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

      9. distribute-rgt-out 99.2%

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

      10. *-commutative 99.2%

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

      11. count-2 99.2%

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

      12. distribute-lft1-in 99.2%

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

      13. metadata-eval 99.2%

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

      14. *-commutative 99.2%

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

      15. *-commutative 99.2%

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

      16. associate-*r* 99.3%

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

      17. cube-unmult 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x.re around 0 99.2%

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

   5. Taylor expanded in x.re around inf 79.9%

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

      1. unpow2 79.9%

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

      2. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if 2.32000000000000015e-14 < x.re < 1.36e103

   1. Initial program 74.8%

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

      1. *-commutative 74.8%

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

      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-+ 51.1%

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

      8. *-commutative 51.1%

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

      9. distribute-lft-in 51.1%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 71.1%

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

   3. Applied egg-rr 71.1%

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

      1. count-2 71.1%

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

      2. distribute-rgt-out 71.1%

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

   5. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + 2\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.9 \cdot 10^{-28}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 2.25 \cdot 10^{-48}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.re\right) - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 2.32 \cdot 10^{-14}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.36 \cdot 10^{+103}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re - x.im \cdot x.im\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.re\right) - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.18 \cdot 10^{-17}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.36 \cdot 10^{+103}:\\ \;\;\;\;x.im \cdot \left(2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* 3.0 (* x.re (* x.im x.re)))))
   (if (<= x.re -1.8e-26)
     t_0
     (if (<= x.re 3.2e-48)
       (* x.im (- (* x.re (- x.re)) (* x.im x.im)))
       (if (<= x.re 1.18e-17)
         (* 3.0 (* x.im (* x.re x.re)))
         (if (<= x.re 1.36e+103) (* x.im (- 2.0 (* x.im x.im))) t_0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	double tmp;
	if (x_46_re <= -1.8e-26) {
		tmp = t_0;
	} else if (x_46_re <= 3.2e-48) {
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	} else if (x_46_re <= 1.18e-17) {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_re <= 1.36e+103) {
		tmp = x_46_im * (2.0 - (x_46_im * x_46_im));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (x_46re * (x_46im * x_46re))
    if (x_46re <= (-1.8d-26)) then
        tmp = t_0
    else if (x_46re <= 3.2d-48) then
        tmp = x_46im * ((x_46re * -x_46re) - (x_46im * x_46im))
    else if (x_46re <= 1.18d-17) then
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    else if (x_46re <= 1.36d+103) then
        tmp = x_46im * (2.0d0 - (x_46im * x_46im))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	double tmp;
	if (x_46_re <= -1.8e-26) {
		tmp = t_0;
	} else if (x_46_re <= 3.2e-48) {
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	} else if (x_46_re <= 1.18e-17) {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_re <= 1.36e+103) {
		tmp = x_46_im * (2.0 - (x_46_im * x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re))
	tmp = 0
	if x_46_re <= -1.8e-26:
		tmp = t_0
	elif x_46_re <= 3.2e-48:
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im))
	elif x_46_re <= 1.18e-17:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	elif x_46_re <= 1.36e+103:
		tmp = x_46_im * (2.0 - (x_46_im * x_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_re)))
	tmp = 0.0
	if (x_46_re <= -1.8e-26)
		tmp = t_0;
	elseif (x_46_re <= 3.2e-48)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * Float64(-x_46_re)) - Float64(x_46_im * x_46_im)));
	elseif (x_46_re <= 1.18e-17)
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	elseif (x_46_re <= 1.36e+103)
		tmp = Float64(x_46_im * Float64(2.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = 3.0 * (x_46_re * (x_46_im * x_46_re));
	tmp = 0.0;
	if (x_46_re <= -1.8e-26)
		tmp = t_0;
	elseif (x_46_re <= 3.2e-48)
		tmp = x_46_im * ((x_46_re * -x_46_re) - (x_46_im * x_46_im));
	elseif (x_46_re <= 1.18e-17)
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	elseif (x_46_re <= 1.36e+103)
		tmp = x_46_im * (2.0 - (x_46_im * x_46_im));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.8e-26], t$95$0, If[LessEqual[x$46$re, 3.2e-48], N[(x$46$im * N[(N[(x$46$re * (-x$46$re)), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.18e-17], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.36e+103], N[(x$46$im * N[(2.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.8000000000000001e-26 or 1.36e103 < x.re

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. *-commutative 71.3%

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

      3. sub-neg 71.3%

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

      4. distribute-lft-in 65.6%

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

      5. associate-+r+ 65.6%

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

      6. distribute-rgt-neg-out 65.6%

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

      7. unsub-neg 65.6%

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

      8. associate-*r* 76.0%

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

      9. distribute-rgt-out 76.0%

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

      10. *-commutative 76.0%

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

      11. count-2 76.0%

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

      12. distribute-lft1-in 76.0%

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

      13. metadata-eval 76.0%

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

      14. *-commutative 76.0%

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

      15. *-commutative 76.0%

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

      16. associate-*r* 76.0%

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

      17. cube-unmult 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x.re around 0 76.0%

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

   5. Taylor expanded in x.re around inf 76.0%

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

      1. unpow2 76.0%

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

      2. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   9. Applied egg-rr 86.5%

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

   if -1.8000000000000001e-26 < x.re < 3.1999999999999998e-48

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. associate-+r+ 99.9%

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

      6. distribute-rgt-neg-out 99.9%

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

      7. unsub-neg 99.9%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 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* 99.9%

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

      17. cube-unmult 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x.re around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-neg 100.0%

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

      4. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. unsub-neg 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. unpow3 99.8%

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

      3. distribute-rgt-out-- 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   10. Applied egg-rr 99.9%

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

   12. Simplified 97.9%

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

   if 3.1999999999999998e-48 < x.re < 1.18000000000000004e-17

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. associate-+r+ 99.4%

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

      6. distribute-rgt-neg-out 99.4%

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

      7. unsub-neg 99.4%

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

      8. associate-*r* 99.0%

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

      9. distribute-rgt-out 99.2%

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

      10. *-commutative 99.2%

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

      11. count-2 99.2%

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

      12. distribute-lft1-in 99.2%

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

      13. metadata-eval 99.2%

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

      14. *-commutative 99.2%

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

      15. *-commutative 99.2%

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

      16. associate-*r* 99.3%

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

      17. cube-unmult 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x.re around 0 99.2%

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

   5. Taylor expanded in x.re around inf 79.9%

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

      1. unpow2 79.9%

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

      2. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if 1.18000000000000004e-17 < x.re < 1.36e103

   1. Initial program 74.8%

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

      1. *-commutative 74.8%

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

      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-+ 51.1%

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

      8. *-commutative 51.1%

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

      9. distribute-lft-in 51.1%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 71.1%

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

   3. Applied egg-rr 71.1%

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

      1. count-2 71.1%

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

      2. distribute-rgt-out 71.1%

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

   5. Applied egg-rr 71.1%

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

   6. Taylor expanded in x.re around 0 67.2%

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

      1. *-commutative 67.2%

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

      2. unpow2 67.2%

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

   8. Simplified 67.2%

      \[\leadsto \color{blue}{x.im \cdot \left(2 - x.im \cdot x.im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot \left(-x.re\right) - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.re \leq 1.18 \cdot 10^{-17}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \mathbf{elif}\;x.re \leq 1.36 \cdot 10^{+103}:\\ \;\;\;\;x.im \cdot \left(2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 6: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+153} \lor \neg \left(x.re \leq 4.2 \cdot 10^{+116}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -7.5e+153) (not (<= x.re 4.2e+116)))
   (* 3.0 (* x.re (* x.im x.re)))
   (* x.im (- (* 3.0 (* x.re x.re)) (* x.im x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7.5e+153) || !(x_46_re <= 4.2e+116)) {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	} else {
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-7.5d+153)) .or. (.not. (x_46re <= 4.2d+116))) then
        tmp = 3.0d0 * (x_46re * (x_46im * x_46re))
    else
        tmp = x_46im * ((3.0d0 * (x_46re * x_46re)) - (x_46im * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -7.5e+153) || !(x_46_re <= 4.2e+116)) {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	} else {
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -7.5e+153) or not (x_46_re <= 4.2e+116):
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re))
	else:
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -7.5e+153) || !(x_46_re <= 4.2e+116))
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_re)));
	else
		tmp = Float64(x_46_im * Float64(Float64(3.0 * Float64(x_46_re * x_46_re)) - Float64(x_46_im * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -7.5e+153) || ~((x_46_re <= 4.2e+116)))
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	else
		tmp = x_46_im * ((3.0 * (x_46_re * x_46_re)) - (x_46_im * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -7.5e+153], N[Not[LessEqual[x$46$re, 4.2e+116]], $MachinePrecision]], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(3.0 * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -7.50000000000000065e153 or 4.2000000000000002e116 < x.re

   1. Initial program 63.1%

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

      1. +-commutative 63.1%

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

      2. *-commutative 63.1%

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

      3. sub-neg 63.1%

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

      4. distribute-lft-in 55.4%

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

      5. associate-+r+ 55.4%

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

      6. distribute-rgt-neg-out 55.4%

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

      7. unsub-neg 55.4%

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

      8. associate-*r* 72.2%

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

      9. distribute-rgt-out 72.2%

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

      10. *-commutative 72.2%

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

      11. count-2 72.2%

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

      12. distribute-lft1-in 72.2%

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

      13. metadata-eval 72.2%

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

      14. *-commutative 72.2%

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

      15. *-commutative 72.2%

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

      16. associate-*r* 72.2%

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

      17. cube-unmult 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x.re around 0 72.2%

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

   5. Taylor expanded in x.re around inf 75.3%

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

      1. unpow2 75.3%

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

      2. *-commutative 75.3%

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

   7. Simplified 75.3%

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

   9. Applied egg-rr 92.2%

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

   if -7.50000000000000065e153 < x.re < 4.2000000000000002e116

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. *-commutative 94.0%

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

      3. sub-neg 94.0%

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

      4. distribute-lft-in 93.5%

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

      5. associate-+r+ 93.5%

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

      6. distribute-rgt-neg-out 93.5%

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

      7. unsub-neg 93.5%

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

      8. associate-*r* 93.5%

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

      9. distribute-rgt-out 93.5%

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

      10. *-commutative 93.5%

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

      11. count-2 93.5%

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

      12. distribute-lft1-in 93.5%

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

      13. metadata-eval 93.5%

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

      14. *-commutative 93.5%

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

      15. *-commutative 93.5%

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

      16. associate-*r* 93.5%

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

      17. cube-unmult 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in x.re around 0 93.6%

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

      1. associate-*r* 93.6%

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

      2. *-commutative 93.6%

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

      3. fma-neg 94.6%

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

      4. *-commutative 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. fma-udef 93.6%

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

      2. unsub-neg 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r* 93.6%

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

      2. unpow3 93.5%

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

      3. distribute-rgt-out-- 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.8%

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

   10. Applied egg-rr 99.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.5 \cdot 10^{+153} \lor \neg \left(x.re \leq 4.2 \cdot 10^{+116}\right):\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(3 \cdot \left(x.re \cdot x.re\right) - x.im \cdot x.im\right)\\ \end{array} \]

Alternative 7: 24.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.3 \cdot 10^{-297}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{elif}\;x.re \leq 4.4 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (- x.im))))
   (if (<= x.re -2.55e+210)
     t_0
     (if (<= x.re -2.3e-297)
       (* x.im x.re)
       (if (<= x.re 4.4e+179) t_0 (* (* x.im x.re) 2.0))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * -x_46_im;
	double tmp;
	if (x_46_re <= -2.55e+210) {
		tmp = t_0;
	} else if (x_46_re <= -2.3e-297) {
		tmp = x_46_im * x_46_re;
	} else if (x_46_re <= 4.4e+179) {
		tmp = t_0;
	} else {
		tmp = (x_46_im * x_46_re) * 2.0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x_46re * -x_46im
    if (x_46re <= (-2.55d+210)) then
        tmp = t_0
    else if (x_46re <= (-2.3d-297)) then
        tmp = x_46im * x_46re
    else if (x_46re <= 4.4d+179) then
        tmp = t_0
    else
        tmp = (x_46im * x_46re) * 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * -x_46_im;
	double tmp;
	if (x_46_re <= -2.55e+210) {
		tmp = t_0;
	} else if (x_46_re <= -2.3e-297) {
		tmp = x_46_im * x_46_re;
	} else if (x_46_re <= 4.4e+179) {
		tmp = t_0;
	} else {
		tmp = (x_46_im * x_46_re) * 2.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * -x_46_im
	tmp = 0
	if x_46_re <= -2.55e+210:
		tmp = t_0
	elif x_46_re <= -2.3e-297:
		tmp = x_46_im * x_46_re
	elif x_46_re <= 4.4e+179:
		tmp = t_0
	else:
		tmp = (x_46_im * x_46_re) * 2.0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(-x_46_im))
	tmp = 0.0
	if (x_46_re <= -2.55e+210)
		tmp = t_0;
	elseif (x_46_re <= -2.3e-297)
		tmp = Float64(x_46_im * x_46_re);
	elseif (x_46_re <= 4.4e+179)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im * x_46_re) * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * -x_46_im;
	tmp = 0.0;
	if (x_46_re <= -2.55e+210)
		tmp = t_0;
	elseif (x_46_re <= -2.3e-297)
		tmp = x_46_im * x_46_re;
	elseif (x_46_re <= 4.4e+179)
		tmp = t_0;
	else
		tmp = (x_46_im * x_46_re) * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * (-x$46$im)), $MachinePrecision]}, If[LessEqual[x$46$re, -2.55e+210], t$95$0, If[LessEqual[x$46$re, -2.3e-297], N[(x$46$im * x$46$re), $MachinePrecision], If[LessEqual[x$46$re, 4.4e+179], t$95$0, N[(N[(x$46$im * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.55e210 or -2.2999999999999999e-297 < x.re < 4.4000000000000001e179

   1. Initial program 84.4%

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

   2. Taylor expanded in x.re around inf 46.3%

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

   4. Simplified 46.3%

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

      1. *-commutative 46.3%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 25.1%

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

      8. *-commutative 25.1%

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

      9. distribute-lft-in 25.1%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in x.re around 0 14.0%

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

   9. Simplified 26.1%

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

   if -2.55e210 < x.re < -2.2999999999999999e-297

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      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-+ 65.1%

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

      8. *-commutative 65.1%

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

      9. distribute-lft-in 65.1%

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

      10. *-un-lft-identity 65.1%

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

      11. fma-def 65.1%

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

   3. Applied egg-rr 65.1%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in x.re around inf 18.4%

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

      1. unpow2 18.4%

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

      2. distribute-rgt-out 18.4%

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

   8. Simplified 18.4%

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

   9. Taylor expanded in x.re around 0 17.7%

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

   if 4.4000000000000001e179 < x.re

   1. Initial program 71.9%

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

   2. Taylor expanded in x.re around inf 93.4%

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

   4. Simplified 93.4%

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

      1. *-commutative 93.4%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 93.4%

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

      8. *-commutative 93.4%

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

      9. distribute-lft-in 93.4%

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

   6. Applied egg-rr 93.4%

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

   7. Taylor expanded in x.re around 0 46.1%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.55 \cdot 10^{+210}:\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{elif}\;x.re \leq -2.3 \cdot 10^{-297}:\\ \;\;\;\;x.im \cdot x.re\\ \mathbf{elif}\;x.re \leq 4.4 \cdot 10^{+179}:\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot x.re\right) \cdot 2\\ \end{array} \]

Alternative 8: 54.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{+150} \lor \neg \left(x.im \leq 5.2 \cdot 10^{+197}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -6e+150) (not (<= x.im 5.2e+197)))
   (* x.re (* x.re (- x.im)))
   (* 3.0 (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6e+150) || !(x_46_im <= 5.2e+197)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-6d+150)) .or. (.not. (x_46im <= 5.2d+197))) then
        tmp = x_46re * (x_46re * -x_46im)
    else
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -6e+150) || !(x_46_im <= 5.2e+197)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -6e+150) or not (x_46_im <= 5.2e+197):
		tmp = x_46_re * (x_46_re * -x_46_im)
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -6e+150) || !(x_46_im <= 5.2e+197))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -6e+150) || ~((x_46_im <= 5.2e+197)))
		tmp = x_46_re * (x_46_re * -x_46_im);
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -6e+150], N[Not[LessEqual[x$46$im, 5.2e+197]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -6.00000000000000025e150 or 5.19999999999999975e197 < x.im

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. *-commutative 57.4%

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

      3. sub-neg 57.4%

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

      4. distribute-lft-in 57.4%

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

      5. associate-+r+ 57.4%

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

      6. distribute-rgt-neg-out 57.4%

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

      7. unsub-neg 57.4%

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

      8. associate-*r* 57.4%

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

      9. distribute-rgt-out 57.4%

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

      10. *-commutative 57.4%

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

      11. count-2 57.4%

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

      12. distribute-lft1-in 57.4%

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

      13. metadata-eval 57.4%

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

      14. *-commutative 57.4%

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

      15. *-commutative 57.4%

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

      16. associate-*r* 57.4%

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

      17. cube-unmult 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in x.re around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. fma-neg 61.7%

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

      4. *-commutative 61.7%

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

   6. Applied egg-rr 61.7%

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

      1. fma-udef 57.4%

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

      2. unsub-neg 57.4%

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

   8. Applied egg-rr 57.4%

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

   9. Taylor expanded in x.re around inf 9.0%

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

   11. Simplified 36.3%

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

   if -6.00000000000000025e150 < x.im < 5.19999999999999975e197

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. *-commutative 92.6%

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

      3. sub-neg 92.6%

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

      4. distribute-lft-in 89.7%

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

      5. associate-+r+ 89.7%

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

      6. distribute-rgt-neg-out 89.7%

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

      7. unsub-neg 89.7%

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

      8. associate-*r* 95.0%

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

      9. distribute-rgt-out 95.0%

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

      10. *-commutative 95.0%

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

      11. count-2 95.0%

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

      12. distribute-lft1-in 95.0%

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

      13. metadata-eval 95.0%

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

      14. *-commutative 95.0%

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

      15. *-commutative 95.0%

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

      16. associate-*r* 95.0%

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

      17. cube-unmult 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x.re around 0 95.0%

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

   5. Taylor expanded in x.re around inf 60.0%

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

      1. unpow2 60.0%

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

      2. *-commutative 60.0%

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

   7. Simplified 60.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6 \cdot 10^{+150} \lor \neg \left(x.im \leq 5.2 \cdot 10^{+197}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 9: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{+29} \lor \neg \left(x.im \leq 1.26 \cdot 10^{+39}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3e+29) (not (<= x.im 1.26e+39)))
   (* x.im (- -2.0 (* x.im x.im)))
   (* 3.0 (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3e+29) || !(x_46_im <= 1.26e+39)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-3d+29)) .or. (.not. (x_46im <= 1.26d+39))) then
        tmp = x_46im * ((-2.0d0) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3e+29) || !(x_46_im <= 1.26e+39)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -3e+29) or not (x_46_im <= 1.26e+39):
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3e+29) || !(x_46_im <= 1.26e+39))
		tmp = Float64(x_46_im * Float64(-2.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3e+29) || ~((x_46_im <= 1.26e+39)))
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -3e+29], N[Not[LessEqual[x$46$im, 1.26e+39]], $MachinePrecision]], N[(x$46$im * N[(-2.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.9999999999999999e29 or 1.26000000000000001e39 < x.im

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      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-+ 78.2%

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

      8. *-commutative 78.2%

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

      9. distribute-lft-in 78.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. count-2 87.7%

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

      2. distribute-rgt-out 87.7%

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

   5. Applied egg-rr 87.7%

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

   6. Taylor expanded in x.re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. unpow2 79.2%

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

   8. Simplified 79.2%

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

      1. expm1-log1p-u 49.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   10. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   12. Simplified 79.2%

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

   if -2.9999999999999999e29 < x.im < 1.26000000000000001e39

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. sub-neg 92.4%

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

      4. distribute-lft-in 92.4%

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

      5. associate-+r+ 92.4%

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

      6. distribute-rgt-neg-out 92.4%

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

      7. unsub-neg 92.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

   5. Taylor expanded in x.re around inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{+29} \lor \neg \left(x.im \leq 1.26 \cdot 10^{+39}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 10: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+27} \lor \neg \left(x.im \leq 2.5 \cdot 10^{+37}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.5e+27) (not (<= x.im 2.5e+37)))
   (* x.im (- -2.0 (* x.im x.im)))
   (* (* x.im 3.0) (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.5e+27) || !(x_46_im <= 2.5e+37)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_im * 3.0) * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.5d+27)) .or. (.not. (x_46im <= 2.5d+37))) then
        tmp = x_46im * ((-2.0d0) - (x_46im * x_46im))
    else
        tmp = (x_46im * 3.0d0) * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.5e+27) || !(x_46_im <= 2.5e+37)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = (x_46_im * 3.0) * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.5e+27) or not (x_46_im <= 2.5e+37):
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im))
	else:
		tmp = (x_46_im * 3.0) * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.5e+27) || !(x_46_im <= 2.5e+37))
		tmp = Float64(x_46_im * Float64(-2.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(Float64(x_46_im * 3.0) * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.5e+27) || ~((x_46_im <= 2.5e+37)))
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	else
		tmp = (x_46_im * 3.0) * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.5e+27], N[Not[LessEqual[x$46$im, 2.5e+37]], $MachinePrecision]], N[(x$46$im * N[(-2.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im * 3.0), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.49999999999999988e27 or 2.49999999999999994e37 < x.im

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      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-+ 78.2%

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

      8. *-commutative 78.2%

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

      9. distribute-lft-in 78.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. count-2 87.7%

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

      2. distribute-rgt-out 87.7%

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

   5. Applied egg-rr 87.7%

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

   6. Taylor expanded in x.re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. unpow2 79.2%

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

   8. Simplified 79.2%

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

      1. expm1-log1p-u 49.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   10. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   12. Simplified 79.2%

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

   if -1.49999999999999988e27 < x.im < 2.49999999999999994e37

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. sub-neg 92.4%

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

      4. distribute-lft-in 92.4%

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

      5. associate-+r+ 92.4%

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

      6. distribute-rgt-neg-out 92.4%

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

      7. unsub-neg 92.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

   5. Taylor expanded in x.re around inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-*r* 71.3%

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

      4. *-commutative 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+27} \lor \neg \left(x.im \leq 2.5 \cdot 10^{+37}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im \cdot 3\right) \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 11: 80.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+27} \lor \neg \left(x.im \leq 9.6 \cdot 10^{+33}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5e+27) (not (<= x.im 9.6e+33)))
   (* x.im (- -2.0 (* x.im x.im)))
   (* 3.0 (* x.re (* x.im x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+27) || !(x_46_im <= 9.6e+33)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-5d+27)) .or. (.not. (x_46im <= 9.6d+33))) then
        tmp = x_46im * ((-2.0d0) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (x_46im * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+27) || !(x_46_im <= 9.6e+33)) {
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5e+27) or not (x_46_im <= 9.6e+33):
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -5e+27) || !(x_46_im <= 9.6e+33))
		tmp = Float64(x_46_im * Float64(-2.0 - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_im * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -5e+27) || ~((x_46_im <= 9.6e+33)))
		tmp = x_46_im * (-2.0 - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_im * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5e+27], N[Not[LessEqual[x$46$im, 9.6e+33]], $MachinePrecision]], N[(x$46$im * N[(-2.0 - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.99999999999999979e27 or 9.5999999999999999e33 < x.im

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      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-+ 78.2%

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

      8. *-commutative 78.2%

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

      9. distribute-lft-in 78.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 87.7%

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

   3. Applied egg-rr 87.7%

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

      1. count-2 87.7%

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

      2. distribute-rgt-out 87.7%

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

   5. Applied egg-rr 87.7%

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

   6. Taylor expanded in x.re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. unpow2 79.2%

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

   8. Simplified 79.2%

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

      1. expm1-log1p-u 49.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)\right)} \]

      2. expm1-udef 49.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   10. Applied egg-rr 49.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x.im \cdot \left(2 - x.im \cdot x.im\right)\right)} - 1} \]

   12. Simplified 79.2%

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

   if -4.99999999999999979e27 < x.im < 9.5999999999999999e33

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. *-commutative 92.4%

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

      3. sub-neg 92.4%

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

      4. distribute-lft-in 92.4%

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

      5. associate-+r+ 92.4%

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

      6. distribute-rgt-neg-out 92.4%

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

      7. unsub-neg 92.4%

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

      8. associate-*r* 99.7%

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

      9. distribute-rgt-out 99.7%

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

      10. *-commutative 99.7%

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

      11. count-2 99.7%

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

      12. distribute-lft1-in 99.7%

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

      13. metadata-eval 99.7%

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

      14. *-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

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

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

   5. Taylor expanded in x.re around inf 71.3%

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

      1. unpow2 71.3%

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

      2. *-commutative 71.3%

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

   7. Simplified 71.3%

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

   9. Applied egg-rr 78.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+27} \lor \neg \left(x.im \leq 9.6 \cdot 10^{+33}\right):\\ \;\;\;\;x.im \cdot \left(-2 - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.im \cdot x.re\right)\right)\\ \end{array} \]

Alternative 12: 24.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+209} \lor \neg \left(x.re \leq -2.3 \cdot 10^{-297}\right) \land x.re \leq 1.65 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.re -2.1e+209)
         (and (not (<= x.re -2.3e-297)) (<= x.re 1.65e+181)))
   (* x.re (- x.im))
   (* x.im x.re)))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.1e+209) || (!(x_46_re <= -2.3e-297) && (x_46_re <= 1.65e+181))) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_im * x_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46re <= (-2.1d+209)) .or. (.not. (x_46re <= (-2.3d-297))) .and. (x_46re <= 1.65d+181)) then
        tmp = x_46re * -x_46im
    else
        tmp = x_46im * x_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_re <= -2.1e+209) || (!(x_46_re <= -2.3e-297) && (x_46_re <= 1.65e+181))) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_im * x_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_re <= -2.1e+209) or (not (x_46_re <= -2.3e-297) and (x_46_re <= 1.65e+181)):
		tmp = x_46_re * -x_46_im
	else:
		tmp = x_46_im * x_46_re
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.1e+209) || (!(x_46_re <= -2.3e-297) && (x_46_re <= 1.65e+181)))
		tmp = Float64(x_46_re * Float64(-x_46_im));
	else
		tmp = Float64(x_46_im * x_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_re <= -2.1e+209) || (~((x_46_re <= -2.3e-297)) && (x_46_re <= 1.65e+181)))
		tmp = x_46_re * -x_46_im;
	else
		tmp = x_46_im * x_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$re, -2.1e+209], And[N[Not[LessEqual[x$46$re, -2.3e-297]], $MachinePrecision], LessEqual[x$46$re, 1.65e+181]]], N[(x$46$re * (-x$46$im)), $MachinePrecision], N[(x$46$im * x$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.1e209 or -2.2999999999999999e-297 < x.re < 1.65000000000000008e181

   1. Initial program 84.4%

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

   2. Taylor expanded in x.re around inf 46.3%

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

   4. Simplified 46.3%

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

      1. *-commutative 46.3%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 25.1%

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

      8. *-commutative 25.1%

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

      9. distribute-lft-in 25.1%

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

   6. Applied egg-rr 25.1%

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

   7. Taylor expanded in x.re around 0 14.0%

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

   9. Simplified 26.1%

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

   if -2.1e209 < x.re < -2.2999999999999999e-297 or 1.65000000000000008e181 < x.re

   1. Initial program 87.8%

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

      1. *-commutative 87.8%

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

      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-+ 66.6%

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

      8. *-commutative 66.6%

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

      9. distribute-lft-in 66.6%

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

      10. *-un-lft-identity 66.6%

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

      11. fma-def 66.6%

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

   3. Applied egg-rr 66.6%

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

   5. Simplified 49.9%

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

   6. Taylor expanded in x.re around inf 34.4%

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

      1. unpow2 34.4%

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

      2. distribute-rgt-out 34.4%

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

   8. Simplified 34.4%

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

   9. Taylor expanded in x.re around 0 23.7%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+209} \lor \neg \left(x.re \leq -2.3 \cdot 10^{-297}\right) \land x.re \leq 1.65 \cdot 10^{+181}:\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot x.re\\ \end{array} \]

Alternative 13: 40.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.05 \cdot 10^{+150} \lor \neg \left(x.im \leq 1.8 \cdot 10^{+194}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -3.05e+150) (not (<= x.im 1.8e+194)))
   (* x.re (* x.re (- x.im)))
   (* x.re (* x.im x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.05e+150) || !(x_46_im <= 1.8e+194)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_im * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-3.05d+150)) .or. (.not. (x_46im <= 1.8d+194))) then
        tmp = x_46re * (x_46re * -x_46im)
    else
        tmp = x_46re * (x_46im * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -3.05e+150) || !(x_46_im <= 1.8e+194)) {
		tmp = x_46_re * (x_46_re * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_im * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -3.05e+150) or not (x_46_im <= 1.8e+194):
		tmp = x_46_re * (x_46_re * -x_46_im)
	else:
		tmp = x_46_re * (x_46_im * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -3.05e+150) || !(x_46_im <= 1.8e+194))
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -3.05e+150) || ~((x_46_im <= 1.8e+194)))
		tmp = x_46_re * (x_46_re * -x_46_im);
	else
		tmp = x_46_re * (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, -3.05e+150], N[Not[LessEqual[x$46$im, 1.8e+194]], $MachinePrecision]], N[(x$46$re * N[(x$46$re * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.05000000000000013e150 or 1.8e194 < x.im

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. *-commutative 57.4%

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

      3. sub-neg 57.4%

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

      4. distribute-lft-in 57.4%

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

      5. associate-+r+ 57.4%

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

      6. distribute-rgt-neg-out 57.4%

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

      7. unsub-neg 57.4%

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

      8. associate-*r* 57.4%

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

      9. distribute-rgt-out 57.4%

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

      10. *-commutative 57.4%

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

      11. count-2 57.4%

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

      12. distribute-lft1-in 57.4%

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

      13. metadata-eval 57.4%

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

      14. *-commutative 57.4%

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

      15. *-commutative 57.4%

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

      16. associate-*r* 57.4%

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

      17. cube-unmult 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in x.re around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. fma-neg 61.7%

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

      4. *-commutative 61.7%

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

   6. Applied egg-rr 61.7%

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

      1. fma-udef 57.4%

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

      2. unsub-neg 57.4%

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

   8. Applied egg-rr 57.4%

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

   9. Taylor expanded in x.re around inf 9.0%

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

   11. Simplified 36.3%

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

   if -3.05000000000000013e150 < x.im < 1.8e194

   1. Initial program 92.6%

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

   2. Taylor expanded in x.re around inf 60.1%

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

   4. Simplified 60.1%

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

      1. add-log-exp 38.1%

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

      2. +-commutative 38.1%

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

      3. exp-sum 38.1%

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

   6. Applied egg-rr 41.4%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.05 \cdot 10^{+150} \lor \neg \left(x.im \leq 1.8 \cdot 10^{+194}\right):\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]

Alternative 14: 36.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.75 \cdot 10^{+156} \lor \neg \left(x.im \leq 5.4 \cdot 10^{+218}\right):\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.75e+156) (not (<= x.im 5.4e+218)))
   (* x.re (- x.im))
   (* x.im (* x.re x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.75e+156) || !(x_46_im <= 5.4e+218)) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_im * (x_46_re * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.75d+156)) .or. (.not. (x_46im <= 5.4d+218))) then
        tmp = x_46re * -x_46im
    else
        tmp = x_46im * (x_46re * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.75e+156) || !(x_46_im <= 5.4e+218)) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_im * (x_46_re * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.75e+156) or not (x_46_im <= 5.4e+218):
		tmp = x_46_re * -x_46_im
	else:
		tmp = x_46_im * (x_46_re * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.75e+156) || !(x_46_im <= 5.4e+218))
		tmp = Float64(x_46_re * Float64(-x_46_im));
	else
		tmp = Float64(x_46_im * Float64(x_46_re * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.75e+156) || ~((x_46_im <= 5.4e+218)))
		tmp = x_46_re * -x_46_im;
	else
		tmp = x_46_im * (x_46_re * x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.75e+156], N[Not[LessEqual[x$46$im, 5.4e+218]], $MachinePrecision]], N[(x$46$re * (-x$46$im)), $MachinePrecision], N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.7500000000000002e156 or 5.40000000000000025e218 < x.im

   1. Initial program 59.1%

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

   2. Taylor expanded in x.re around inf 7.2%

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

   4. Simplified 7.2%

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

      1. *-commutative 7.2%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 3.1%

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

      8. *-commutative 3.1%

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

      9. distribute-lft-in 3.1%

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

   6. Applied egg-rr 3.1%

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

   7. Taylor expanded in x.re around 0 12.6%

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

   9. Simplified 27.1%

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

   if -1.7500000000000002e156 < x.im < 5.40000000000000025e218

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      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-+ 63.6%

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

      8. *-commutative 63.6%

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

      9. distribute-lft-in 63.6%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 44.0%

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

   3. Applied egg-rr 44.0%

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

   4. Taylor expanded in x.re around inf 40.7%

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

      1. unpow2 40.7%

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

      2. *-commutative 40.7%

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

   6. Simplified 40.7%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.75 \cdot 10^{+156} \lor \neg \left(x.im \leq 5.4 \cdot 10^{+218}\right):\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 15: 37.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.6 \cdot 10^{+152} \lor \neg \left(x.im \leq 3.2 \cdot 10^{+217}\right):\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.6e+152) (not (<= x.im 3.2e+217)))
   (* x.re (- x.im))
   (* x.re (* x.im x.re))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.6e+152) || !(x_46_im <= 3.2e+217)) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_re * (x_46_im * x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.6d+152)) .or. (.not. (x_46im <= 3.2d+217))) then
        tmp = x_46re * -x_46im
    else
        tmp = x_46re * (x_46im * x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.6e+152) || !(x_46_im <= 3.2e+217)) {
		tmp = x_46_re * -x_46_im;
	} else {
		tmp = x_46_re * (x_46_im * x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.6e+152) or not (x_46_im <= 3.2e+217):
		tmp = x_46_re * -x_46_im
	else:
		tmp = x_46_re * (x_46_im * x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.6e+152) || !(x_46_im <= 3.2e+217))
		tmp = Float64(x_46_re * Float64(-x_46_im));
	else
		tmp = Float64(x_46_re * Float64(x_46_im * x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.6e+152) || ~((x_46_im <= 3.2e+217)))
		tmp = x_46_re * -x_46_im;
	else
		tmp = x_46_re * (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, -1.6e+152], N[Not[LessEqual[x$46$im, 3.2e+217]], $MachinePrecision]], N[(x$46$re * (-x$46$im)), $MachinePrecision], N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.60000000000000003e152 or 3.2000000000000001e217 < x.im

   1. Initial program 59.1%

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

   2. Taylor expanded in x.re around inf 7.2%

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

   4. Simplified 7.2%

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

      1. *-commutative 7.2%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 3.1%

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

      8. *-commutative 3.1%

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

      9. distribute-lft-in 3.1%

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

   6. Applied egg-rr 3.1%

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

   7. Taylor expanded in x.re around 0 12.6%

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

   9. Simplified 27.1%

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

   if -1.60000000000000003e152 < x.im < 3.2000000000000001e217

   1. Initial program 91.8%

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

   2. Taylor expanded in x.re around inf 59.7%

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

   4. Simplified 59.7%

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

      1. add-log-exp 38.1%

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

      2. +-commutative 38.1%

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

      3. exp-sum 38.1%

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

   6. Applied egg-rr 41.3%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.6 \cdot 10^{+152} \lor \neg \left(x.im \leq 3.2 \cdot 10^{+217}\right):\\ \;\;\;\;x.re \cdot \left(-x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot x.re\right)\\ \end{array} \]

Alternative 16: 20.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   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-+ 64.4%

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

   8. *-commutative 64.4%

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

   9. distribute-lft-in 64.4%

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

   10. *-un-lft-identity 64.4%

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

   11. fma-def 64.4%

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

3. Applied egg-rr 64.4%

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

5. Simplified 45.7%

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

6. Taylor expanded in x.re around inf 29.9%

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

   1. unpow2 29.9%

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

   2. distribute-rgt-out 31.8%

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

8. Simplified 31.8%

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

9. Taylor expanded in x.re around 0 19.0%

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

10. Final simplification 19.0%

    \[\leadsto x.im \cdot x.re \]

Alternative 17: 4.4% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -x_46_im

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. *-commutative 86.2%

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

   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-+ 64.4%

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

   8. *-commutative 64.4%

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

   9. distribute-lft-in 64.4%

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

   10. flip-+ 0.0%

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

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. flip-+ 50.9%

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

3. Applied egg-rr 50.9%

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

   1. count-2 50.9%

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

   2. distribute-rgt-out 50.9%

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

5. Applied egg-rr 50.9%

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

6. Taylor expanded in x.re around 0 36.3%

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

   1. *-commutative 36.3%

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

   2. unpow2 36.3%

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

8. Simplified 36.3%

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

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

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

11. Simplified 4.3%

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

12. Final simplification 4.3%

    \[\leadsto -x.im \]

Alternative 18: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.2%

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

   1. +-commutative 86.2%

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

   2. *-commutative 86.2%

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

   3. sub-neg 86.2%

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

   4. distribute-lft-in 83.8%

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

   5. associate-+r+ 83.8%

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

   6. distribute-rgt-neg-out 83.8%

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

   7. unsub-neg 83.8%

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

   8. associate-*r* 88.1%

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

   9. distribute-rgt-out 88.1%

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

   10. *-commutative 88.1%

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

   11. count-2 88.1%

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

   12. distribute-lft1-in 88.1%

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

   13. metadata-eval 88.1%

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

   14. *-commutative 88.1%

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

   15. *-commutative 88.1%

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

   16. associate-*r* 88.1%

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

   17. cube-unmult 88.1%

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

3. Simplified 88.1%

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

4. Taylor expanded in x.re around 0 58.9%

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

6. Simplified 2.8%

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

7. Final simplification 2.8%

   \[\leadsto -3 \]

Developer target: 91.3% 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 2023193 
(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)))