math.cube on complex, imaginary part

Percentage Accurate: 83.1% → 99.8%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

• 44.6% 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: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -5e+103) (not (<= x.im 2e+101)))
   (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))
   (- (* x.re (* x.im (* x.re 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -5e+103) || !(x_46_im <= 2e+101)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - 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 <= (-5d+103)) .or. (.not. (x_46im <= 2d+101))) then
        tmp = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    else
        tmp = (x_46re * (x_46im * (x_46re * 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 <= -5e+103) || !(x_46_im <= 2e+101)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -5e+103) or not (x_46_im <= 2e+101):
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - 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 <= -5e+103) || !(x_46_im <= 2e+101))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 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 <= -5e+103) || ~((x_46_im <= 2e+101)))
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -5e+103], N[Not[LessEqual[x$46$im, 2e+101]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$im * N[(x$46$re * 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 < -5e103 or 2e101 < x.im

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. fma-def 76.0%

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

      4. *-commutative 76.0%

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

      5. distribute-lft-out 74.7%

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

      6. *-commutative 74.7%

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

   3. Simplified 74.7%

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

      1. fma-udef 69.3%

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

      2. distribute-lft-in 70.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 73.3%

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

      9. distribute-lft-in 74.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 81.3%

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

      16. *-commutative 81.3%

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

      17. difference-of-squares 98.7%

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

      18. associate-*r* 98.7%

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

      19. *-commutative 98.7%

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

   5. Applied egg-rr 98.7%

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

   if -5e103 < x.im < 2e101

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. *-commutative 91.0%

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

      3. distribute-lft-out 91.0%

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

      4. associate-*l* 90.9%

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

      5. *-commutative 90.9%

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

      6. distribute-lft-out 90.9%

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

      7. associate-+r- 90.9%

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

      8. distribute-lft-out-- 90.9%

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

   3. Simplified 91.0%

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

      1. sub-neg 91.0%

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

      2. associate-*l* 91.0%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

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

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*l* 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+103} \lor \neg \left(x.im \leq 2 \cdot 10^{+101}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. *-commutative 93.0%

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

      3. difference-of-squares 93.0%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 18.2%

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

      4. *-commutative 18.2%

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

      5. distribute-lft-out 18.2%

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

      6. *-commutative 18.2%

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

   3. Simplified 18.2%

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

      1. fma-udef 0.0%

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

      2. distribute-lft-in 0.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 31.8%

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

      9. distribute-lft-in 31.8%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 40.9%

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

      16. *-commutative 40.9%

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

      17. difference-of-squares 100.0%

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

      18. associate-*r* 100.0%

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

      19. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.5%

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

Alternative 3: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right) - {x.im}^{3}\\ t_1 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (- (* 3.0 (* (* x.re x.re) x.im)) (pow x.im 3.0)))
        (t_1 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -5e+103)
     t_1
     (if (<= x.im -1.45e-95)
       t_0
       (if (<= x.im 2.6e-151)
         (+ (* x.re (+ (* x.re x.im) (* x.re x.im))) (* x.re (* x.re x.im)))
         (if (<= x.im 2e+101) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (3.0 * ((x_46_re * x_46_re) * x_46_im)) - pow(x_46_im, 3.0);
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5e+103) {
		tmp = t_1;
	} else if (x_46_im <= -1.45e-95) {
		tmp = t_0;
	} else if (x_46_im <= 2.6e-151) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 2e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (3.0d0 * ((x_46re * x_46re) * x_46im)) - (x_46im ** 3.0d0)
    t_1 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-5d+103)) then
        tmp = t_1
    else if (x_46im <= (-1.45d-95)) then
        tmp = t_0
    else if (x_46im <= 2.6d-151) then
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + (x_46re * (x_46re * x_46im))
    else if (x_46im <= 2d+101) then
        tmp = t_0
    else
        tmp = t_1
    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_re) * x_46_im)) - Math.pow(x_46_im, 3.0);
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5e+103) {
		tmp = t_1;
	} else if (x_46_im <= -1.45e-95) {
		tmp = t_0;
	} else if (x_46_im <= 2.6e-151) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 2e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (3.0 * ((x_46_re * x_46_re) * x_46_im)) - math.pow(x_46_im, 3.0)
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -5e+103:
		tmp = t_1
	elif x_46_im <= -1.45e-95:
		tmp = t_0
	elif x_46_im <= 2.6e-151:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 2e+101:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im)) - (x_46_im ^ 3.0))
	t_1 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -5e+103)
		tmp = t_1;
	elseif (x_46_im <= -1.45e-95)
		tmp = t_0;
	elseif (x_46_im <= 2.6e-151)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 2e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (3.0 * ((x_46_re * x_46_re) * x_46_im)) - (x_46_im ^ 3.0);
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -5e+103)
		tmp = t_1;
	elseif (x_46_im <= -1.45e-95)
		tmp = t_0;
	elseif (x_46_im <= 2.6e-151)
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 2e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -5e+103], t$95$1, If[LessEqual[x$46$im, -1.45e-95], t$95$0, If[LessEqual[x$46$im, 2.6e-151], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5e103 or 2e101 < x.im

   1. Initial program 70.7%

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

      1. +-commutative 70.7%

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

      2. *-commutative 70.7%

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

      3. fma-def 76.0%

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

      4. *-commutative 76.0%

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

      5. distribute-lft-out 74.7%

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

      6. *-commutative 74.7%

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

   3. Simplified 74.7%

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

      1. fma-udef 69.3%

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

      2. distribute-lft-in 70.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 73.3%

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

      9. distribute-lft-in 74.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 81.3%

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

      16. *-commutative 81.3%

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

      17. difference-of-squares 98.7%

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

      18. associate-*r* 98.7%

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

      19. *-commutative 98.7%

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

   5. Applied egg-rr 98.7%

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

   if -5e103 < x.im < -1.45000000000000001e-95 or 2.6e-151 < x.im < 2e101

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. distribute-lft-out 98.9%

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

      4. associate-*l* 98.8%

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

      5. *-commutative 98.8%

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

      6. distribute-lft-out 98.8%

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

      7. associate-+r- 98.8%

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

      8. distribute-lft-out-- 98.8%

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

   3. Simplified 98.9%

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

   if -1.45000000000000001e-95 < x.im < 2.6e-151

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. sub-neg 79.9%

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

      3. distribute-rgt-in 79.9%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. unpow3 99.8%

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

   3. Applied egg-rr 99.8%

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

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

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

      1. unpow2 79.9%

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

   6. Simplified 79.9%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right) - {x.im}^{3}\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-151}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+101}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right) - {x.im}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := t_0 + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ t_2 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;t_0 + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im) (* x.re x.im))))
        (t_1 (+ t_0 (* x.im (- (* x.re x.re) (* x.im x.im)))))
        (t_2 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -5e+148)
     t_2
     (if (<= x.im -1.45e-95)
       t_1
       (if (<= x.im 7.2e-152)
         (+ t_0 (* x.re (* x.re x.im)))
         (if (<= x.im 2e+63) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	double t_2 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5e+148) {
		tmp = t_2;
	} else if (x_46_im <= -1.45e-95) {
		tmp = t_1;
	} else if (x_46_im <= 7.2e-152) {
		tmp = t_0 + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 2e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x_46re * ((x_46re * x_46im) + (x_46re * x_46im))
    t_1 = t_0 + (x_46im * ((x_46re * x_46re) - (x_46im * x_46im)))
    t_2 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-5d+148)) then
        tmp = t_2
    else if (x_46im <= (-1.45d-95)) then
        tmp = t_1
    else if (x_46im <= 7.2d-152) then
        tmp = t_0 + (x_46re * (x_46re * x_46im))
    else if (x_46im <= 2d+63) then
        tmp = t_1
    else
        tmp = t_2
    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_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	double t_2 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5e+148) {
		tmp = t_2;
	} else if (x_46_im <= -1.45e-95) {
		tmp = t_1;
	} else if (x_46_im <= 7.2e-152) {
		tmp = t_0 + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 2e+63) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)))
	t_2 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -5e+148:
		tmp = t_2
	elif x_46_im <= -1.45e-95:
		tmp = t_1
	elif x_46_im <= 7.2e-152:
		tmp = t_0 + (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 2e+63:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	t_1 = Float64(t_0 + Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))))
	t_2 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -5e+148)
		tmp = t_2;
	elseif (x_46_im <= -1.45e-95)
		tmp = t_1;
	elseif (x_46_im <= 7.2e-152)
		tmp = Float64(t_0 + Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 2e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	t_1 = t_0 + (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im)));
	t_2 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -5e+148)
		tmp = t_2;
	elseif (x_46_im <= -1.45e-95)
		tmp = t_1;
	elseif (x_46_im <= 7.2e-152)
		tmp = t_0 + (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 2e+63)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -5e+148], t$95$2, If[LessEqual[x$46$im, -1.45e-95], t$95$1, If[LessEqual[x$46$im, 7.2e-152], N[(t$95$0 + N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2e+63], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5.00000000000000024e148 or 2.00000000000000012e63 < x.im

   1. Initial program 72.1%

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

      1. +-commutative 72.1%

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

      2. *-commutative 72.1%

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

      3. fma-def 77.2%

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

      4. *-commutative 77.2%

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

      5. distribute-lft-out 75.9%

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

      6. *-commutative 75.9%

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

   3. Simplified 75.9%

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

      1. fma-udef 70.8%

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

      2. distribute-lft-in 72.1%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 73.4%

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

      9. distribute-lft-in 74.6%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 82.2%

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

      16. *-commutative 82.2%

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

      17. difference-of-squares 98.7%

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

      18. associate-*r* 98.7%

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

      19. *-commutative 98.7%

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

   5. Applied egg-rr 98.7%

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

   if -5.00000000000000024e148 < x.im < -1.45000000000000001e-95 or 7.2e-152 < x.im < 2.00000000000000012e63

   1. Initial program 98.9%

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

   if -1.45000000000000001e-95 < x.im < 7.2e-152

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. sub-neg 79.6%

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

      3. distribute-rgt-in 79.6%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. unpow3 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x.re around inf 79.6%

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

      1. unpow2 79.6%

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

   6. Simplified 79.6%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{+148}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{-152}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{+63}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 5: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1250:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 400000000000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -1250.0)
     t_0
     (if (<= x.im 1.8e-112)
       (* 3.0 (* x.re (* x.re x.im)))
       (if (<= x.im 400000000000.0)
         (* x.im (- (* x.re x.re) (* x.im x.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1250.0) {
		tmp = t_0;
	} else if (x_46_im <= 1.8e-112) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 400000000000.0) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (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 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-1250.0d0)) then
        tmp = t_0
    else if (x_46im <= 1.8d-112) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46im <= 400000000000.0d0) then
        tmp = x_46im * ((x_46re * x_46re) - (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 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -1250.0) {
		tmp = t_0;
	} else if (x_46_im <= 1.8e-112) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 400000000000.0) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -1250.0:
		tmp = t_0
	elif x_46_im <= 1.8e-112:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 400000000000.0:
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -1250.0)
		tmp = t_0;
	elseif (x_46_im <= 1.8e-112)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 400000000000.0)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - 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 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -1250.0)
		tmp = t_0;
	elseif (x_46_im <= 1.8e-112)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 400000000000.0)
		tmp = x_46_im * ((x_46_re * x_46_re) - (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[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1250.0], t$95$0, If[LessEqual[x$46$im, 1.8e-112], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 400000000000.0], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1250 or 4e11 < x.im

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. fma-def 85.8%

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

      4. *-commutative 85.8%

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

      5. distribute-lft-out 85.0%

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

      6. *-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. fma-udef 81.9%

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

      2. distribute-lft-in 82.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 77.4%

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

      9. distribute-lft-in 78.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 84.2%

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

      16. *-commutative 84.2%

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

      17. difference-of-squares 94.4%

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

      18. associate-*r* 94.4%

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

      19. *-commutative 94.4%

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

   5. Applied egg-rr 94.4%

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

   if -1250 < x.im < 1.8e-112

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. sub-neg 85.7%

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

      3. distribute-rgt-in 85.7%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. unpow3 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around inf 77.5%

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

      1. unpow2 77.5%

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

      2. *-commutative 77.5%

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

      3. distribute-rgt1-in 77.5%

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

      4. metadata-eval 77.5%

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

      5. associate-*l* 77.4%

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

   8. Simplified 77.4%

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

   10. Applied egg-rr 91.7%

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

   if 1.8e-112 < x.im < 4e11

   1. Initial program 95.5%

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

      1. *-commutative 95.5%

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

      2. sub-neg 95.5%

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

      3. distribute-rgt-in 95.5%

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

      4. associate-*l* 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. unpow3 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   7. Applied egg-rr 73.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1250:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 400000000000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 6: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -5500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 400000000000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -5500.0)
     t_0
     (if (<= x.im 1.8e-112)
       (+ (* x.re (+ (* x.re x.im) (* x.re x.im))) (* x.re (* x.re x.im)))
       (if (<= x.im 400000000000.0)
         (* x.im (- (* x.re x.re) (* x.im x.im)))
         t_0)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5500.0) {
		tmp = t_0;
	} else if (x_46_im <= 1.8e-112) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 400000000000.0) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (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 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-5500.0d0)) then
        tmp = t_0
    else if (x_46im <= 1.8d-112) then
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + (x_46re * (x_46re * x_46im))
    else if (x_46im <= 400000000000.0d0) then
        tmp = x_46im * ((x_46re * x_46re) - (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 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -5500.0) {
		tmp = t_0;
	} else if (x_46_im <= 1.8e-112) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 400000000000.0) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -5500.0:
		tmp = t_0
	elif x_46_im <= 1.8e-112:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 400000000000.0:
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -5500.0)
		tmp = t_0;
	elseif (x_46_im <= 1.8e-112)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 400000000000.0)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - 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 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -5500.0)
		tmp = t_0;
	elseif (x_46_im <= 1.8e-112)
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 400000000000.0)
		tmp = x_46_im * ((x_46_re * x_46_re) - (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[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -5500.0], t$95$0, If[LessEqual[x$46$im, 1.8e-112], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 400000000000.0], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5500 or 4e11 < x.im

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. *-commutative 82.7%

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

      3. fma-def 85.8%

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

      4. *-commutative 85.8%

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

      5. distribute-lft-out 85.0%

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

      6. *-commutative 85.0%

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

   3. Simplified 85.0%

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

      1. fma-udef 81.9%

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

      2. distribute-lft-in 82.7%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 77.4%

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

      9. distribute-lft-in 78.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 84.2%

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

      16. *-commutative 84.2%

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

      17. difference-of-squares 94.4%

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

      18. associate-*r* 94.4%

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

      19. *-commutative 94.4%

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

   5. Applied egg-rr 94.4%

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

   if -5500 < x.im < 1.8e-112

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. sub-neg 85.7%

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

      3. distribute-rgt-in 85.7%

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

      4. associate-*l* 99.8%

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

      5. distribute-lft-neg-in 99.8%

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

      6. unpow3 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x.re around inf 77.6%

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

      1. unpow2 77.6%

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

   6. Simplified 77.6%

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

   8. Applied egg-rr 91.7%

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

   if 1.8e-112 < x.im < 4e11

   1. Initial program 95.5%

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

      1. *-commutative 95.5%

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

      2. sub-neg 95.5%

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

      3. distribute-rgt-in 95.5%

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

      4. associate-*l* 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. unpow3 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

   5. Applied egg-rr 99.7%

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

   7. Applied egg-rr 73.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5500:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 400000000000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 7: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.7e+58)
   (* x.im (- (* x.re x.re) (* x.im x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.7e+58) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 1.7d+58) then
        tmp = x_46im * ((x_46re * x_46re) - (x_46im * x_46im))
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.7e+58) {
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.7e+58:
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.7e+58)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.7e+58)
		tmp = x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im));
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.7e+58], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.7e58

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

      2. sub-neg 90.4%

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

      3. distribute-rgt-in 88.0%

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

      4. associate-*l* 91.1%

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

      5. distribute-lft-neg-in 91.1%

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

      6. unpow3 91.2%

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

   3. Applied egg-rr 91.2%

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

      1. *-commutative 91.2%

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

      2. distribute-lft-out 90.8%

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

   5. Applied egg-rr 90.8%

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

   7. Applied egg-rr 75.0%

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

   if 1.7e58 < x.re

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. sub-neg 61.8%

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

      3. distribute-rgt-in 57.7%

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

      4. associate-*l* 77.0%

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

      5. distribute-lft-neg-in 77.0%

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

      6. unpow3 77.0%

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

   3. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. distribute-lft-out 77.0%

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

   5. Applied egg-rr 77.0%

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

   6. Taylor expanded in x.re around inf 66.0%

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

      1. unpow2 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-rgt1-in 66.0%

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

      4. metadata-eval 66.0%

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

      5. associate-*l* 66.0%

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

   8. Simplified 66.0%

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

   10. Applied egg-rr 85.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 8: 67.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.5e+58)
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.5e+58) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 1.5d+58) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.5e+58) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.5e+58:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.5e+58)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.5e+58)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.5e+58], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.5000000000000001e58

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 63.3%

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

      8. *-commutative 63.3%

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

      9. distribute-lft-in 63.7%

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

   3. Applied egg-rr 63.7%

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

   4. Taylor expanded in x.re around 0 59.7%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + 2 \cdot \left(x.im \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 59.7%

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

      2. *-commutative 59.7%

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

      3. count-2 59.7%

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

      4. distribute-rgt-in 59.7%

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

      5. mul-1-neg 59.7%

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

      6. unsub-neg 59.7%

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

   6. Simplified 59.7%

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

   8. Applied egg-rr 66.8%

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

   if 1.5000000000000001e58 < x.re

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. sub-neg 61.8%

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

      3. distribute-rgt-in 57.7%

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

      4. associate-*l* 77.0%

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

      5. distribute-lft-neg-in 77.0%

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

      6. unpow3 77.0%

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

   3. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. distribute-lft-out 77.0%

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

   5. Applied egg-rr 77.0%

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

   6. Taylor expanded in x.re around inf 66.0%

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

      1. unpow2 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-rgt1-in 66.0%

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

      4. metadata-eval 66.0%

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

      5. associate-*l* 66.0%

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

   8. Simplified 66.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 9: 70.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1.55 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 1.55e+58)
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.re (* x.re x.im)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.55e+58) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 1.55d+58) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 1.55e+58) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 1.55e+58:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 1.55e+58)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 1.55e+58)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 1.55e+58], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.55e58

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 63.3%

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

      8. *-commutative 63.3%

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

      9. distribute-lft-in 63.7%

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

   3. Applied egg-rr 63.7%

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

   4. Taylor expanded in x.re around 0 59.7%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + 2 \cdot \left(x.im \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 59.7%

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

      2. *-commutative 59.7%

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

      3. count-2 59.7%

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

      4. distribute-rgt-in 59.7%

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

      5. mul-1-neg 59.7%

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

      6. unsub-neg 59.7%

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

   6. Simplified 59.7%

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

   8. Applied egg-rr 66.8%

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

   if 1.55e58 < x.re

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

      2. sub-neg 61.8%

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

      3. distribute-rgt-in 57.7%

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

      4. associate-*l* 77.0%

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

      5. distribute-lft-neg-in 77.0%

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

      6. unpow3 77.0%

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

   3. Applied egg-rr 77.0%

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

      1. *-commutative 77.0%

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

      2. distribute-lft-out 77.0%

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

   5. Applied egg-rr 77.0%

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

   6. Taylor expanded in x.re around inf 66.0%

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

      1. unpow2 66.0%

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

      2. *-commutative 66.0%

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

      3. distribute-rgt1-in 66.0%

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

      4. metadata-eval 66.0%

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

      5. associate-*l* 66.0%

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

   8. Simplified 66.0%

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

   10. Applied egg-rr 85.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1.55 \cdot 10^{+58}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \end{array} \]

Alternative 10: 65.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.4 \cdot 10^{+100}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.re 2.4e+100) (* x.im (* x.im (- x.im))) (* x.re (* x.re x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.4e+100) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46re <= 2.4d+100) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = x_46re * (x_46re * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_re <= 2.4e+100) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_re <= 2.4e+100:
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = x_46_re * (x_46_re * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_re <= 2.4e+100)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_re <= 2.4e+100)
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = x_46_re * (x_46_re * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$re, 2.4e+100], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.40000000000000012e100

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      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-+ 62.5%

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

      8. *-commutative 62.5%

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

      9. distribute-lft-in 62.9%

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

   3. Applied egg-rr 62.9%

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

   4. Taylor expanded in x.re around 0 58.7%

      \[\leadsto \color{blue}{-1 \cdot {x.im}^{3} + 2 \cdot \left(x.im \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. +-commutative 58.7%

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

      2. *-commutative 58.7%

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

      3. count-2 58.7%

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

      4. distribute-rgt-in 58.7%

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

      5. mul-1-neg 58.7%

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

      6. unsub-neg 58.7%

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

   6. Simplified 58.7%

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

   8. Applied egg-rr 66.1%

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

   if 2.40000000000000012e100 < x.re

   1. Initial program 60.2%

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

      1. *-commutative 60.2%

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

      2. sub-neg 60.2%

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

      3. distribute-rgt-in 55.3%

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

      4. associate-*l* 77.9%

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

      5. distribute-lft-neg-in 77.9%

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

      6. unpow3 77.9%

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

   3. Applied egg-rr 77.9%

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

   4. Taylor expanded in x.re around inf 67.5%

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

      1. unpow2 67.5%

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

   6. Simplified 67.5%

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

   8. Applied egg-rr 62.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.4 \cdot 10^{+100}:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im\right)\\ \end{array} \]

Alternative 11: 35.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

   1. *-commutative 85.0%

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

   2. sub-neg 85.0%

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

   3. distribute-rgt-in 82.3%

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

   4. associate-*l* 88.5%

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

   5. distribute-lft-neg-in 88.5%

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

   6. unpow3 88.6%

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

3. Applied egg-rr 88.6%

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

4. Taylor expanded in x.re around inf 50.2%

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

   1. unpow2 50.2%

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

6. Simplified 50.2%

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

8. Applied egg-rr 34.5%

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

9. Final simplification 34.5%

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

Alternative 12: 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 85.0%

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

2. Taylor expanded in x.re around 0 66.8%

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

4. Simplified 21.5%

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

   1. *-commutative 21.5%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

      \[\leadsto -3 + \frac{\color{blue}{0}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

   4. +-inverses 0.0%

      \[\leadsto -3 + \frac{\color{blue}{x.im \cdot x.im - x.im \cdot x.im}}{x.re \cdot x.im - x.re \cdot x.im} \cdot x.re \]

   5. +-inverses 0.0%

      \[\leadsto -3 + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}} \cdot x.re \]

   6. +-inverses 0.0%

      \[\leadsto -3 + \frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{x.im - x.im}} \cdot x.re \]

   7. flip-+ 8.2%

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

   8. *-commutative 8.2%

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

   9. distribute-rgt-in 8.2%

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

   10. *-commutative 8.2%

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

   11. flip-+ 0.0%

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

   12. clear-num 0.0%

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

   13. *-commutative 0.0%

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

   14. +-inverses 0.0%

       \[\leadsto -3 + \frac{1}{\frac{\color{blue}{0}}{\left(x.re \cdot x.im\right) \cdot \left(x.re \cdot x.im\right) - \left(x.im \cdot x.re\right) \cdot \left(x.im \cdot x.re\right)}} \]

   15. +-inverses 0.0%

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

   16. *-commutative 0.0%

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

   17. *-commutative 0.0%

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

   18. +-inverses 0.0%

       \[\leadsto -3 + \frac{1}{\frac{x.im \cdot x.im - x.im \cdot x.im}{\color{blue}{0}}} \]

   19. +-inverses 0.0%

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

   20. flip-+ 2.8%

       \[\leadsto -3 + \frac{1}{\color{blue}{x.im + x.im}} \]

6. Applied egg-rr 2.8%

   \[\leadsto -3 + \color{blue}{\frac{1}{x.im + x.im}} \]

7. Taylor expanded in x.im around inf 2.7%

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

8. Final simplification 2.7%

   \[\leadsto -3 \]

Alternative 13: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -2.6 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -2.6)

C

double code(double x_46_re, double x_46_im) {
	return -2.6;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -2.6d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -2.6;
}

Python

def code(x_46_re, x_46_im):
	return -2.6

Julia

function code(x_46_re, x_46_im)
	return -2.6
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -2.6;
end

Wolfram

code[x$46$re_, x$46$im_] := -2.6

Derivation

1. Initial program 85.0%

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

   1. +-commutative 85.0%

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

   2. *-commutative 85.0%

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

   3. distribute-lft-out 85.0%

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

   4. associate-*l* 85.0%

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

   5. *-commutative 85.0%

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

   6. distribute-lft-out 88.5%

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

   7. associate-+r- 88.5%

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

   8. distribute-lft-out-- 82.2%

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

3. Simplified 82.3%

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

   1. flip3-- 11.7%

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

   2. frac-2neg 11.7%

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

   3. *-commutative 11.7%

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

   4. unpow-prod-down 11.6%

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

   5. metadata-eval 11.6%

      \[\leadsto \frac{-\left(\color{blue}{27} \cdot {\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}^{3} - {\left({x.im}^{3}\right)}^{3}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   6. associate-*l* 11.6%

      \[\leadsto \frac{-\left(27 \cdot {\color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}}^{3} - {\left({x.im}^{3}\right)}^{3}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   7. pow-pow 11.6%

      \[\leadsto \frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - \color{blue}{{x.im}^{\left(3 \cdot 3\right)}}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

   8. metadata-eval 11.6%

      \[\leadsto \frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{\color{blue}{9}}\right)}{-\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)\right)} \]

5. Applied egg-rr 8.1%

   \[\leadsto \color{blue}{\frac{-\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{9}\right)}{-\left(\left(9 \cdot \left(\left(x.im \cdot x.im\right) \cdot {x.re}^{4}\right) + {x.im}^{6}\right) + \left(x.re \cdot x.re\right) \cdot \left(\left(x.im \cdot 3\right) \cdot {x.im}^{3}\right)\right)}} \]

7. Simplified 2.7%

   \[\leadsto \color{blue}{-2.6} \]

8. Final simplification 2.7%

   \[\leadsto -2.6 \]

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 2023271 
(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)))