math.cube on complex, imaginary part

Percentage Accurate: 82.6% → 99.2%

Time: 7.8s

Alternatives: 13

Speedup: 1.0×

• 44.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.5e+117) (not (<= x.im 1.5e+80)))
   (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))
   (- (* (* 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 <= -1.5e+117) || !(x_46_im <= 1.5e+80)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= (-1.5d+117)) .or. (.not. (x_46im <= 1.5d+80))) then
        tmp = (x_46im + x_46im) + (x_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    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 <= -1.5e+117) || !(x_46_im <= 1.5e+80)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= -1.5e+117) or not (x_46_im <= 1.5e+80):
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	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 <= -1.5e+117) || !(x_46_im <= 1.5e+80))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))));
	else
		tmp = Float64(Float64(Float64(x_46_re * Float64(x_46_im * 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 <= -1.5e+117) || ~((x_46_im <= 1.5e+80)))
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	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, -1.5e+117], N[Not[LessEqual[x$46$im, 1.5e+80]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * N[(x$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.5e117 or 1.49999999999999993e80 < x.im

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-def 77.8%

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

      4. *-commutative 77.8%

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

      5. distribute-rgt-out 77.8%

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

      6. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 71.6%

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

      2. distribute-lft-in 71.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.5e117 < x.im < 1.49999999999999993e80

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. *-commutative 92.2%

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

      3. sub-neg 92.2%

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

      4. distribute-lft-in 92.2%

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

      5. associate-+r+ 92.2%

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

      6. distribute-rgt-neg-out 92.2%

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

      7. unsub-neg 92.2%

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

   3. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.re \cdot \left(x.im \cdot x.re\right)\right) \cdot 3 - {x.im}^{3}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 2 \cdot 10^{+79}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.5e+117) (not (<= x.im 2e+79)))
   (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))
   (- (* 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 <= -1.5e+117) || !(x_46_im <= 2e+79)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= (-1.5d+117)) .or. (.not. (x_46im <= 2d+79))) then
        tmp = (x_46im + x_46im) + (x_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    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 <= -1.5e+117) || !(x_46_im <= 2e+79)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= -1.5e+117) or not (x_46_im <= 2e+79):
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	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 <= -1.5e+117) || !(x_46_im <= 2e+79))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im * 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 <= -1.5e+117) || ~((x_46_im <= 2e+79)))
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	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, -1.5e+117], N[Not[LessEqual[x$46$im, 2e+79]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.5e117 or 1.99999999999999993e79 < x.im

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-def 77.8%

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

      4. *-commutative 77.8%

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

      5. distribute-rgt-out 77.8%

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

      6. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 71.6%

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

      2. distribute-lft-in 71.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.5e117 < x.im < 1.99999999999999993e79

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. *-commutative 92.2%

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

      3. sub-neg 92.2%

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

      4. distribute-lft-in 92.2%

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

      5. associate-+r+ 92.2%

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

      6. distribute-rgt-neg-out 92.2%

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

      7. unsub-neg 92.2%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.6%

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

      10. *-commutative 99.6%

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

      11. count-2 99.6%

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

      12. distribute-lft1-in 99.6%

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

      13. metadata-eval 99.6%

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

      14. *-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around 0 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 2 \cdot 10^{+79}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 3\right) - {x.im}^{3}\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 2 \cdot 10^{+79}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.5e+117) (not (<= x.im 2e+79)))
   (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))
   (- (* 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 <= -1.5e+117) || !(x_46_im <= 2e+79)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= (-1.5d+117)) .or. (.not. (x_46im <= 2d+79))) then
        tmp = (x_46im + x_46im) + (x_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    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 <= -1.5e+117) || !(x_46_im <= 2e+79)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} 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 <= -1.5e+117) or not (x_46_im <= 2e+79):
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	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 <= -1.5e+117) || !(x_46_im <= 2e+79))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_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 <= -1.5e+117) || ~((x_46_im <= 2e+79)))
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	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, -1.5e+117], N[Not[LessEqual[x$46$im, 2e+79]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $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 < -1.5e117 or 1.99999999999999993e79 < x.im

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-def 77.8%

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

      4. *-commutative 77.8%

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

      5. distribute-rgt-out 77.8%

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

      6. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 71.6%

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

      2. distribute-lft-in 71.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.5e117 < x.im < 1.99999999999999993e79

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. *-commutative 92.2%

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

      3. sub-neg 92.2%

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

      4. distribute-lft-in 92.2%

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

      5. associate-+r+ 92.2%

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

      6. distribute-rgt-neg-out 92.2%

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

      7. unsub-neg 92.2%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.6%

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

      10. *-commutative 99.6%

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

      11. count-2 99.6%

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

      12. distribute-lft1-in 99.6%

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

      13. metadata-eval 99.6%

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

      14. *-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x.re around 0 99.7%

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

      1. expm1-log1p-u 85.1%

         \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

      2. expm1-udef 54.7%

         \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]

   6. Applied egg-rr 54.7%

      \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]
   7. Step-by-step derivation

      1. expm1-def 85.1%

         \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l* 99.7%

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

      6. *-commutative 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 2 \cdot 10^{+79}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.im \cdot \left(x.re \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -1.5e+117) (not (<= x.im 1.5e+80)))
   (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))
   (- (* x.re (* x.re (* x.im 3.0))) (pow x.im 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.5e+117) || !(x_46_im <= 1.5e+80)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - pow(x_46_im, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-1.5d+117)) .or. (.not. (x_46im <= 1.5d+80))) then
        tmp = (x_46im + x_46im) + (x_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    else
        tmp = (x_46re * (x_46re * (x_46im * 3.0d0))) - (x_46im ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -1.5e+117) || !(x_46_im <= 1.5e+80)) {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	} else {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -1.5e+117) or not (x_46_im <= 1.5e+80):
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	else:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -1.5e+117) || !(x_46_im <= 1.5e+80))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))));
	else
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -1.5e+117) || ~((x_46_im <= 1.5e+80)))
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	else
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -1.5e+117], N[Not[LessEqual[x$46$im, 1.5e+80]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.5e117 or 1.49999999999999993e80 < x.im

   1. Initial program 71.6%

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

      1. +-commutative 71.6%

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

      2. *-commutative 71.6%

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

      3. fma-def 77.8%

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

      4. *-commutative 77.8%

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

      5. distribute-rgt-out 77.8%

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

      6. *-commutative 77.8%

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

   3. Simplified 77.8%

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

      1. fma-udef 71.6%

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

      2. distribute-lft-in 71.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.5e117 < x.im < 1.49999999999999993e80

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. *-commutative 92.2%

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

      3. sub-neg 92.2%

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

      4. distribute-lft-in 92.2%

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

      5. associate-+r+ 92.2%

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

      6. distribute-rgt-neg-out 92.2%

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

      7. unsub-neg 92.2%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.6%

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

      10. *-commutative 99.6%

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

      11. count-2 99.6%

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

      12. distribute-lft1-in 99.6%

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

      13. metadata-eval 99.6%

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

      14. *-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.5 \cdot 10^{+117} \lor \neg \left(x.im \leq 1.5 \cdot 10^{+80}\right):\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right) - {x.im}^{3}\\ \end{array} \]

Alternative 5: 94.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_im * x_46_re) + (x_46_im * x_46_re)));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   if +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 21.7%

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

      4. *-commutative 21.7%

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

      5. distribute-rgt-out 21.7%

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

      6. *-commutative 21.7%

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

   3. Simplified 21.7%

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

      1. fma-udef 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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 47.8%

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

   5. Applied egg-rr 47.8%

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

      1. difference-of-squares 100.0%

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

      2. *-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 94.7%

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

Alternative 6: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{if}\;x.im \leq -56000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{-40}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 6200:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))))
   (if (<= x.im -56000000.0)
     t_0
     (if (<= x.im 7.2e-40)
       (+ (* x.re (* (* x.im x.re) 2.0)) (* x.im (* x.re x.re)))
       (if (<= x.im 6200.0) (* x.im (* 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_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	double tmp;
	if (x_46_im <= -56000000.0) {
		tmp = t_0;
	} else if (x_46_im <= 7.2e-40) {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_im <= 6200.0) {
		tmp = x_46_im * (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_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    if (x_46im <= (-56000000.0d0)) then
        tmp = t_0
    else if (x_46im <= 7.2d-40) then
        tmp = (x_46re * ((x_46im * x_46re) * 2.0d0)) + (x_46im * (x_46re * x_46re))
    else if (x_46im <= 6200.0d0) then
        tmp = x_46im * (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_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	double tmp;
	if (x_46_im <= -56000000.0) {
		tmp = t_0;
	} else if (x_46_im <= 7.2e-40) {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_im <= 6200.0) {
		tmp = x_46_im * (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_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	tmp = 0
	if x_46_im <= -56000000.0:
		tmp = t_0
	elif x_46_im <= 7.2e-40:
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re))
	elif x_46_im <= 6200.0:
		tmp = x_46_im * (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(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))))
	tmp = 0.0
	if (x_46_im <= -56000000.0)
		tmp = t_0;
	elseif (x_46_im <= 7.2e-40)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) * 2.0)) + Float64(x_46_im * Float64(x_46_re * x_46_re)));
	elseif (x_46_im <= 6200.0)
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-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_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -56000000.0)
		tmp = t_0;
	elseif (x_46_im <= 7.2e-40)
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	elseif (x_46_im <= 6200.0)
		tmp = x_46_im * (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[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -56000000.0], t$95$0, If[LessEqual[x$46$im, 7.2e-40], N[(N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6200.0], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5.6e7 or 6200 < x.im

   1. Initial program 80.6%

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

      1. +-commutative 80.6%

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

      2. *-commutative 80.6%

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

      3. fma-def 84.8%

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

      4. *-commutative 84.8%

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

      5. distribute-rgt-out 84.8%

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

      6. *-commutative 84.8%

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

   3. Simplified 84.8%

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

      1. fma-udef 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 86.7%

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

   5. Applied egg-rr 86.7%

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

      1. difference-of-squares 96.8%

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

      2. *-commutative 96.8%

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

   7. Applied egg-rr 96.8%

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

   if -5.6e7 < x.im < 7.2e-40

   1. Initial program 89.5%

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

   2. Taylor expanded in x.re around inf 83.6%

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

   4. Simplified 83.6%

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

   5. Taylor expanded in x.re around 0 83.6%

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

   if 7.2e-40 < x.im < 6200

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. flip-+ 99.2%

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

      3. pow2 99.2%

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

      4. pow2 99.2%

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

      5. pow-prod-up 99.2%

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

      6. metadata-eval 99.2%

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

      7. distribute-rgt-neg-in 99.2%

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

      8. distribute-rgt-neg-in 99.2%

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

      9. distribute-rgt-neg-in 99.2%

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

   3. Applied egg-rr 99.2%

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

   4. Taylor expanded in x.re around 0 83.1%

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

      1. unpow2 83.1%

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

      2. neg-mul-1 83.1%

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

   6. Simplified 83.1%

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

      1. +-commutative 83.1%

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

      2. *-commutative 83.1%

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

   8. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{\left(x.im \cdot -1\right) \cdot \left(x.im \cdot x.im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -56000000:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{-40}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 6200:\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right)\\ t_1 := \left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{if}\;x.im \leq -155000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;t_0 + x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+55}:\\ \;\;\;\;t_0 - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (* (* x.im x.re) 2.0)))
        (t_1 (+ (+ x.im x.im) (* x.im (* (- x.re x.im) (+ x.im x.re))))))
   (if (<= x.im -155000000.0)
     t_1
     (if (<= x.im 7.5e-64)
       (+ t_0 (* x.im (* x.re x.re)))
       (if (<= x.im 7.6e+55) (- t_0 (* x.im (* x.im x.im))) t_1)))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_im * x_46_re) * 2.0);
	double t_1 = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	double tmp;
	if (x_46_im <= -155000000.0) {
		tmp = t_1;
	} else if (x_46_im <= 7.5e-64) {
		tmp = t_0 + (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_im <= 7.6e+55) {
		tmp = t_0 - (x_46_im * (x_46_im * x_46_im));
	} 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 = x_46re * ((x_46im * x_46re) * 2.0d0)
    t_1 = (x_46im + x_46im) + (x_46im * ((x_46re - x_46im) * (x_46im + x_46re)))
    if (x_46im <= (-155000000.0d0)) then
        tmp = t_1
    else if (x_46im <= 7.5d-64) then
        tmp = t_0 + (x_46im * (x_46re * x_46re))
    else if (x_46im <= 7.6d+55) then
        tmp = t_0 - (x_46im * (x_46im * x_46im))
    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 = x_46_re * ((x_46_im * x_46_re) * 2.0);
	double t_1 = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	double tmp;
	if (x_46_im <= -155000000.0) {
		tmp = t_1;
	} else if (x_46_im <= 7.5e-64) {
		tmp = t_0 + (x_46_im * (x_46_re * x_46_re));
	} else if (x_46_im <= 7.6e+55) {
		tmp = t_0 - (x_46_im * (x_46_im * x_46_im));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_im * x_46_re) * 2.0)
	t_1 = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)))
	tmp = 0
	if x_46_im <= -155000000.0:
		tmp = t_1
	elif x_46_im <= 7.5e-64:
		tmp = t_0 + (x_46_im * (x_46_re * x_46_re))
	elif x_46_im <= 7.6e+55:
		tmp = t_0 - (x_46_im * (x_46_im * x_46_im))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) * 2.0))
	t_1 = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(Float64(x_46_re - x_46_im) * Float64(x_46_im + x_46_re))))
	tmp = 0.0
	if (x_46_im <= -155000000.0)
		tmp = t_1;
	elseif (x_46_im <= 7.5e-64)
		tmp = Float64(t_0 + Float64(x_46_im * Float64(x_46_re * x_46_re)));
	elseif (x_46_im <= 7.6e+55)
		tmp = Float64(t_0 - Float64(x_46_im * Float64(x_46_im * x_46_im)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_im * x_46_re) * 2.0);
	t_1 = (x_46_im + x_46_im) + (x_46_im * ((x_46_re - x_46_im) * (x_46_im + x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -155000000.0)
		tmp = t_1;
	elseif (x_46_im <= 7.5e-64)
		tmp = t_0 + (x_46_im * (x_46_re * x_46_re));
	elseif (x_46_im <= 7.6e+55)
		tmp = t_0 - (x_46_im * (x_46_im * x_46_im));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(N[(x$46$re - x$46$im), $MachinePrecision] * N[(x$46$im + x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -155000000.0], t$95$1, If[LessEqual[x$46$im, 7.5e-64], N[(t$95$0 + N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 7.6e+55], N[(t$95$0 - N[(x$46$im * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.55e8 or 7.5999999999999999e55 < x.im

   1. Initial program 79.4%

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

      1. +-commutative 79.4%

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

      2. *-commutative 79.4%

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

      3. fma-def 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt-out 83.9%

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

      6. *-commutative 83.9%

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

   3. Simplified 83.9%

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

      1. fma-udef 79.4%

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

      2. distribute-lft-in 79.4%

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

      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%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. difference-of-squares 98.5%

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

      2. *-commutative 98.5%

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

   7. Applied egg-rr 98.5%

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

   if -1.55e8 < x.im < 7.49999999999999949e-64

   1. Initial program 89.0%

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

   2. Taylor expanded in x.re around inf 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in x.re around 0 84.4%

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

   if 7.49999999999999949e-64 < x.im < 7.5999999999999999e55

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. flip-+ 71.9%

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

      3. pow2 71.9%

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

      4. pow2 71.9%

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

      5. pow-prod-up 71.9%

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

      6. metadata-eval 71.9%

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

      7. distribute-rgt-neg-in 71.9%

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

      8. distribute-rgt-neg-in 71.9%

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

      9. distribute-rgt-neg-in 71.9%

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

   3. Applied egg-rr 71.9%

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

   4. Taylor expanded in x.re around 0 80.0%

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

      1. unpow2 80.0%

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

      2. neg-mul-1 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in x.re around 0 80.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -155000000:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \mathbf{elif}\;x.im \leq 7.5 \cdot 10^{-64}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \mathbf{elif}\;x.im \leq 7.6 \cdot 10^{+55}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) - x.im \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\\ \end{array} \]

Alternative 8: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.1 \cdot 10^{+96} \lor \neg \left(x.im \leq 1.55 \cdot 10^{-39}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -4.1e+96) (not (<= x.im 1.55e-39)))
   (* x.im (* x.im (- x.im)))
   (+ (* x.re (* (* x.im x.re) 2.0)) (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4.1e+96) || !(x_46_im <= 1.55e-39)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-4.1d+96)) .or. (.not. (x_46im <= 1.55d-39))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = (x_46re * ((x_46im * x_46re) * 2.0d0)) + (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -4.1e+96) || !(x_46_im <= 1.55e-39)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -4.1e+96) or not (x_46_im <= 1.55e-39):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -4.1e+96) || !(x_46_im <= 1.55e-39))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_im * x_46_re) * 2.0)) + Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -4.1e+96) || ~((x_46_im <= 1.55e-39)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = (x_46_re * ((x_46_im * x_46_re) * 2.0)) + (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -4.1e+96], N[Not[LessEqual[x$46$im, 1.55e-39]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$im * x$46$re), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.09999999999999998e96 or 1.54999999999999985e-39 < x.im

   1. Initial program 79.0%

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

      1. sub-neg 79.0%

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

      2. flip-+ 28.1%

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

      3. pow2 28.1%

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

      4. pow2 28.1%

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

      5. pow-prod-up 28.1%

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

      6. metadata-eval 28.1%

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

      7. distribute-rgt-neg-in 28.1%

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

      8. distribute-rgt-neg-in 28.1%

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

      9. distribute-rgt-neg-in 28.1%

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

   3. Applied egg-rr 28.1%

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

   4. Taylor expanded in x.re around 0 69.6%

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

      1. unpow2 69.6%

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

      2. neg-mul-1 69.6%

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

   6. Simplified 69.6%

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

      1. +-commutative 69.6%

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

      2. *-commutative 69.6%

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

   8. Applied egg-rr 79.9%

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

   if -4.09999999999999998e96 < x.im < 1.54999999999999985e-39

   1. Initial program 90.7%

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

   2. Taylor expanded in x.re around inf 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in x.re around 0 80.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.1 \cdot 10^{+96} \lor \neg \left(x.im \leq 1.55 \cdot 10^{-39}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(\left(x.im \cdot x.re\right) \cdot 2\right) + x.im \cdot \left(x.re \cdot x.re\right)\\ \end{array} \]

Alternative 9: 72.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+96} \lor \neg \left(x.im \leq 4.9 \cdot 10^{-40}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -2.4e+96) (not (<= x.im 4.9e-40)))
   (* x.im (* x.im (- x.im)))
   (* 3.0 (* x.im (* x.re x.re)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+96) || !(x_46_im <= 4.9e-40)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-2.4d+96)) .or. (.not. (x_46im <= 4.9d-40))) then
        tmp = x_46im * (x_46im * -x_46im)
    else
        tmp = 3.0d0 * (x_46im * (x_46re * x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -2.4e+96) || !(x_46_im <= 4.9e-40)) {
		tmp = x_46_im * (x_46_im * -x_46_im);
	} else {
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -2.4e+96) or not (x_46_im <= 4.9e-40):
		tmp = x_46_im * (x_46_im * -x_46_im)
	else:
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -2.4e+96) || !(x_46_im <= 4.9e-40))
		tmp = Float64(x_46_im * Float64(x_46_im * Float64(-x_46_im)));
	else
		tmp = Float64(3.0 * Float64(x_46_im * Float64(x_46_re * x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -2.4e+96) || ~((x_46_im <= 4.9e-40)))
		tmp = x_46_im * (x_46_im * -x_46_im);
	else
		tmp = 3.0 * (x_46_im * (x_46_re * x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -2.4e+96], N[Not[LessEqual[x$46$im, 4.9e-40]], $MachinePrecision]], N[(x$46$im * N[(x$46$im * (-x$46$im)), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(x$46$im * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.39999999999999993e96 or 4.8999999999999997e-40 < x.im

   1. Initial program 79.0%

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

      1. sub-neg 79.0%

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

      2. flip-+ 28.1%

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

      3. pow2 28.1%

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

      4. pow2 28.1%

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

      5. pow-prod-up 28.1%

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

      6. metadata-eval 28.1%

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

      7. distribute-rgt-neg-in 28.1%

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

      8. distribute-rgt-neg-in 28.1%

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

      9. distribute-rgt-neg-in 28.1%

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

   3. Applied egg-rr 28.1%

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

   4. Taylor expanded in x.re around 0 69.6%

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

      1. unpow2 69.6%

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

      2. neg-mul-1 69.6%

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

   6. Simplified 69.6%

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

      1. +-commutative 69.6%

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

      2. *-commutative 69.6%

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

   8. Applied egg-rr 79.9%

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

   if -2.39999999999999993e96 < x.im < 4.8999999999999997e-40

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. sub-neg 90.7%

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

      4. distribute-lft-in 90.7%

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

      5. associate-+r+ 90.7%

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

      6. distribute-rgt-neg-out 90.7%

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

      7. unsub-neg 90.7%

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

      8. associate-*r* 99.8%

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

      9. distribute-rgt-out 99.6%

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

      10. *-commutative 99.6%

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

      11. count-2 99.6%

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

      12. distribute-lft1-in 99.6%

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

      13. metadata-eval 99.6%

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

      14. *-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. associate-*r* 99.7%

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

      17. cube-unmult 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x.re around 0 99.6%

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

      1. expm1-log1p-u 85.1%

         \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

      2. expm1-udef 49.0%

         \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]

   6. Applied egg-rr 49.0%

      \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]
   7. Step-by-step derivation

      1. expm1-def 85.1%

         \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

      2. expm1-log1p 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l* 99.7%

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

      6. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. fma-neg 99.7%

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

      3. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x.re around inf 80.8%

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

       1. *-commutative 80.8%

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

       2. unpow2 80.8%

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

   13. Simplified 80.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.4 \cdot 10^{+96} \lor \neg \left(x.im \leq 4.9 \cdot 10^{-40}\right):\\ \;\;\;\;x.im \cdot \left(x.im \cdot \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right)\\ \end{array} \]

Alternative 10: 50.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. *-commutative 85.7%

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

   3. sub-neg 85.7%

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

   4. distribute-lft-in 83.0%

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

   5. associate-+r+ 83.0%

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

   6. distribute-rgt-neg-out 83.0%

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

   7. unsub-neg 83.0%

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

   8. associate-*r* 88.1%

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

   9. distribute-rgt-out 88.0%

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

   10. *-commutative 88.0%

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

   11. count-2 88.0%

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

   12. distribute-lft1-in 88.0%

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

   13. metadata-eval 88.0%

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

   14. *-commutative 88.0%

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

   15. *-commutative 88.0%

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

   16. associate-*r* 88.1%

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

   17. cube-unmult 88.1%

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

3. Simplified 88.1%

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

4. Taylor expanded in x.re around 0 88.0%

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

   1. expm1-log1p-u 71.5%

      \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

   2. expm1-udef 50.7%

      \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]

6. Applied egg-rr 50.7%

   \[\leadsto x.re \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)} - 1\right)} - {x.im}^{3} \]
7. Step-by-step derivation

   1. expm1-def 71.5%

      \[\leadsto x.re \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \left(x.re \cdot x.im\right)\right)\right)} - {x.im}^{3} \]

   2. expm1-log1p 88.0%

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

   3. *-commutative 88.0%

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

   4. *-commutative 88.0%

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

   5. associate-*l* 88.1%

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

   6. *-commutative 88.1%

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

8. Simplified 88.1%

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

   1. associate-*r* 88.1%

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

   2. fma-neg 90.0%

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

   3. *-commutative 90.0%

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

10. Applied egg-rr 90.0%

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

11. Taylor expanded in x.re around inf 54.5%

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

    1. *-commutative 54.5%

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

    2. unpow2 54.5%

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

13. Simplified 54.5%

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

14. Final simplification 54.5%

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

Alternative 11: 20.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 85.7%

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

   2. flip-+ 37.3%

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

   3. pow2 37.3%

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

   4. pow2 37.3%

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

   5. pow-prod-up 37.3%

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

   6. metadata-eval 37.3%

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

   7. distribute-rgt-neg-in 37.3%

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

   8. distribute-rgt-neg-in 37.3%

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

   9. distribute-rgt-neg-in 37.3%

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

3. Applied egg-rr 37.3%

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

4. Taylor expanded in x.re around 0 64.7%

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

   1. unpow2 64.7%

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

   2. neg-mul-1 64.7%

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

6. Simplified 64.7%

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

   1. +-commutative 64.7%

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

   2. *-commutative 64.7%

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

8. Applied egg-rr 18.7%

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

9. Final simplification 18.7%

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

Alternative 12: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -10.0

Julia

function code(x_46_re, x_46_im)
	return -10.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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%

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

   2. *-commutative 85.7%

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

   3. sub-neg 85.7%

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

   4. distribute-lft-in 83.0%

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

   5. associate-+r+ 83.0%

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

   6. distribute-rgt-neg-out 83.0%

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

   7. unsub-neg 83.0%

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

   8. associate-*r* 88.1%

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

   9. distribute-rgt-out 88.0%

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

   10. *-commutative 88.0%

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

   11. count-2 88.0%

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

   12. distribute-lft1-in 88.0%

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

   13. metadata-eval 88.0%

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

   14. *-commutative 88.0%

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

   15. *-commutative 88.0%

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

   16. associate-*r* 88.1%

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

   17. cube-unmult 88.1%

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

3. Simplified 88.1%

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

   1. associate-*r* 88.0%

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

   2. associate-*l* 88.2%

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

   3. flip-- 23.9%

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

   4. div-inv 23.8%

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

   5. swap-sqr 23.8%

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

   6. pow2 23.8%

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

   7. metadata-eval 23.8%

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

   8. pow-prod-up 23.7%

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

   9. metadata-eval 23.7%

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

   10. associate-*l* 23.7%

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

   11. associate-*r* 23.7%

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

   12. fma-def 23.7%

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

5. Applied egg-rr 23.7%

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

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto -10 \]

Alternative 13: 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 85.7%

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

2. Taylor expanded in x.re around 0 52.7%

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

4. Simplified 2.8%

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

5. Final simplification 2.8%

   \[\leadsto -3 \]

Developer target: 91.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023201 
(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)))