math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 99.8%

Time: 11.4s

Alternatives: 13

Speedup: 1.1×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

      2. *-commutative 93.9%

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

      3. difference-of-squares 93.9%

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

      4. associate-*l* 99.7%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 12.1%

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

      4. *-commutative 12.1%

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

      5. distribute-lft-out 12.1%

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

      6. *-commutative 12.1%

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

   3. Simplified 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im + x.im\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   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-+ 42.4%

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

      17. *-commutative 42.4%

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

      18. difference-of-squares 100.0%

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

      19. associate-*r* 100.0%

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

      20. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im))))))
   (if (<= t_0 5e+246)
     t_0
     (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im)))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (t_0 <= 5e+246) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46re * x_46im) + (x_46re * x_46im)))
    if (t_0 <= 5d+246) then
        tmp = t_0
    else
        tmp = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double tmp;
	if (t_0 <= 5e+246) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	tmp = 0
	if t_0 <= 5e+246:
		tmp = t_0
	else:
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))))
	tmp = 0.0
	if (t_0 <= 5e+246)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	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_re * x_46_im) + (x_46_re * x_46_im)));
	tmp = 0.0;
	if (t_0 <= 5e+246)
		tmp = t_0;
	else
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+246], t$95$0, N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

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)) < 4.99999999999999976e246

   1. Initial program 95.1%

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

   if 4.99999999999999976e246 < (+.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 58.9%

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

      1. +-commutative 58.9%

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

      2. *-commutative 58.9%

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

      3. fma-def 63.1%

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

      4. *-commutative 63.1%

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

      5. distribute-lft-out 63.1%

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

      6. *-commutative 63.1%

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

   3. Simplified 63.1%

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

      1. fma-udef 58.9%

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

      2. distribute-lft-in 58.9%

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

      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-+ 73.8%

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

      17. *-commutative 73.8%

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

      18. difference-of-squares 94.0%

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

      19. associate-*r* 94.7%

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

      20. *-commutative 94.7%

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

   5. Applied egg-rr 94.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 5 \cdot 10^{+246}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ t_1 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + t_1 \leq \infty:\\ \;\;\;\;t_1 + t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* (+ x.re x.im) (* x.im (- x.re x.im))))
        (t_1 (* x.re (+ (* x.re x.im) (* x.re x.im)))))
   (if (<= (+ (* x.im (- (* x.re x.re) (* x.im x.im))) t_1) INFINITY)
     (+ t_1 t_0)
     (+ (+ x.im x.im) t_0))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im));
	double t_1 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_1) <= ((double) INFINITY)) {
		tmp = t_1 + t_0;
	} else {
		tmp = (x_46_im + x_46_im) + t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im));
	double t_1 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_1) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 + t_0;
	} else {
		tmp = (x_46_im + x_46_im) + t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im))
	t_1 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_1) <= math.inf:
		tmp = t_1 + t_0
	else:
		tmp = (x_46_im + x_46_im) + t_0
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im)))
	t_1 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + t_1) <= Inf)
		tmp = Float64(t_1 + t_0);
	else
		tmp = Float64(Float64(x_46_im + x_46_im) + t_0);
	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_im * (x_46_re - x_46_im));
	t_1 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_1) <= Inf)
		tmp = t_1 + t_0;
	else
		tmp = (x_46_im + x_46_im) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(t$95$1 + t$95$0), $MachinePrecision], N[(N[(x$46$im + x$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 93.9%

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

      1. difference-of-squares 93.9%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. flip-+ 93.8%

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

      5. associate-*l/ 90.7%

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

   3. Applied egg-rr 90.7%

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

      1. *-un-lft-identity 90.7%

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

      2. associate-/l* 93.8%

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

   5. Applied egg-rr 93.8%

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

      1. associate-/r/ 93.8%

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

      2. flip-+ 99.7%

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

   7. Applied egg-rr 99.7%

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

   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 12.1%

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

      4. *-commutative 12.1%

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

      5. distribute-lft-out 12.1%

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

      6. *-commutative 12.1%

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

   3. Simplified 12.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im + x.im\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   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-+ 42.4%

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

      17. *-commutative 42.4%

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

      18. difference-of-squares 100.0%

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

      19. associate-*r* 100.0%

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

      20. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -3.5 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -2.8 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;x.im \leq 5000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))))
        (t_1 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -3.5e+155)
     t_1
     (if (<= x.im -2.8e-23)
       t_0
       (if (<= x.im 1.6e-64)
         (* x.im (* (* x.re x.re) 3.0))
         (if (<= x.im 5000000.0) t_0 t_1))))))

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));
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -3.5e+155) {
		tmp = t_1;
	} else if (x_46_im <= -2.8e-23) {
		tmp = t_0;
	} else if (x_46_im <= 1.6e-64) {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	} else if (x_46_im <= 5000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + ((x_46re * x_46im) + (x_46re * x_46im))
    t_1 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-3.5d+155)) then
        tmp = t_1
    else if (x_46im <= (-2.8d-23)) then
        tmp = t_0
    else if (x_46im <= 1.6d-64) then
        tmp = x_46im * ((x_46re * x_46re) * 3.0d0)
    else if (x_46im <= 5000000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (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));
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -3.5e+155) {
		tmp = t_1;
	} else if (x_46_im <= -2.8e-23) {
		tmp = t_0;
	} else if (x_46_im <= 1.6e-64) {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	} else if (x_46_im <= 5000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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))
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -3.5e+155:
		tmp = t_1
	elif x_46_im <= -2.8e-23:
		tmp = t_0
	elif x_46_im <= 1.6e-64:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	elif x_46_im <= 5000000.0:
		tmp = t_0
	else:
		tmp = t_1
	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(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	t_1 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -3.5e+155)
		tmp = t_1;
	elseif (x_46_im <= -2.8e-23)
		tmp = t_0;
	elseif (x_46_im <= 1.6e-64)
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	elseif (x_46_im <= 5000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	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));
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -3.5e+155)
		tmp = t_1;
	elseif (x_46_im <= -2.8e-23)
		tmp = t_0;
	elseif (x_46_im <= 1.6e-64)
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	elseif (x_46_im <= 5000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -3.5e+155], t$95$1, If[LessEqual[x$46$im, -2.8e-23], t$95$0, If[LessEqual[x$46$im, 1.6e-64], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5000000.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.49999999999999985e155 or 5e6 < x.im

   1. Initial program 67.0%

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

      1. +-commutative 67.0%

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

      2. *-commutative 67.0%

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

      3. fma-def 71.0%

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

      4. *-commutative 71.0%

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

      5. distribute-lft-out 71.0%

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

      6. *-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. fma-udef 67.0%

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

      2. distribute-lft-in 67.0%

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

      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-+ 78.4%

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

      17. *-commutative 78.4%

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

      18. difference-of-squares 97.4%

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

      19. associate-*r* 97.4%

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

      20. *-commutative 97.4%

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

   5. Applied egg-rr 97.4%

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

   if -3.49999999999999985e155 < x.im < -2.7999999999999997e-23 or 1.59999999999999988e-64 < x.im < 5e6

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 87.0%

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

   3. Applied egg-rr 87.0%

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

   if -2.7999999999999997e-23 < x.im < 1.59999999999999988e-64

   1. Initial program 87.6%

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

      1. +-commutative 87.6%

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

      2. *-commutative 87.6%

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

      3. distribute-lft-out 87.6%

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

      4. associate-*l* 87.6%

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

      5. *-commutative 87.6%

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

      6. distribute-lft-out 87.7%

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

      7. associate-+r- 87.7%

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

      8. distribute-lft-out-- 87.7%

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

   3. Simplified 87.7%

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

      1. sub-neg 87.7%

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

      2. associate-*l* 87.7%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around inf 82.5%

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

      1. *-commutative 82.5%

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

      2. *-commutative 82.5%

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

      3. associate-*l* 82.6%

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

      4. *-commutative 82.6%

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

      5. unpow2 82.6%

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

   8. Simplified 82.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{+155}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -2.8 \cdot 10^{-23}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{-64}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \mathbf{elif}\;x.im \leq 5000000:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{+155}:\\ \;\;\;\;\left(x.im + x.im\right) + x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -30 \lor \neg \left(x.im \leq 2.45 \cdot 10^{+31}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -3.5e+155)
   (+ (+ x.im x.im) (* x.re (* x.im x.im)))
   (if (or (<= x.im -30.0) (not (<= x.im 2.45e+31)))
     (+ (* x.im (- (* x.re x.re) (* x.im x.im))) (+ x.im x.im))
     (* x.im (* (* x.re x.re) 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -3.5e+155) {
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	} else if ((x_46_im <= -30.0) || !(x_46_im <= 2.45e+31)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-3.5d+155)) then
        tmp = (x_46im + x_46im) + (x_46re * (x_46im * x_46im))
    else if ((x_46im <= (-30.0d0)) .or. (.not. (x_46im <= 2.45d+31))) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46im + x_46im)
    else
        tmp = x_46im * ((x_46re * x_46re) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -3.5e+155) {
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	} else if ((x_46_im <= -30.0) || !(x_46_im <= 2.45e+31)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -3.5e+155:
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im))
	elif (x_46_im <= -30.0) or not (x_46_im <= 2.45e+31):
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im)
	else:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -3.5e+155)
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_re * Float64(x_46_im * x_46_im)));
	elseif ((x_46_im <= -30.0) || !(x_46_im <= 2.45e+31))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_im + x_46_im));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -3.5e+155)
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	elseif ((x_46_im <= -30.0) || ~((x_46_im <= 2.45e+31)))
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	else
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -3.5e+155], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$im, -30.0], N[Not[LessEqual[x$46$im, 2.45e+31]], $MachinePrecision]], 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$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.49999999999999985e155

   1. Initial program 55.6%

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

      1. +-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. fma-def 61.1%

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

      4. *-commutative 61.1%

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

      5. distribute-lft-out 61.1%

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

      6. *-commutative 61.1%

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

   3. Simplified 61.1%

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

      1. fma-udef 55.6%

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

      2. distribute-lft-in 55.6%

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

      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-+ 69.4%

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

      17. *-commutative 69.4%

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

      18. difference-of-squares 100.0%

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

      19. associate-*r* 100.0%

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

      20. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 50.7%

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

   7. Taylor expanded in x.re around 0 52.8%

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

      1. unpow2 52.8%

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

   9. Simplified 52.8%

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

   if -3.49999999999999985e155 < x.im < -30 or 2.44999999999999998e31 < x.im

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. *-commutative 80.4%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. associate-*r/ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. distribute-lft-out-- 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. +-inverses 0.0%

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

      16. flip-+ 89.3%

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

   3. Applied egg-rr 89.3%

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

   if -30 < x.im < 2.44999999999999998e31

   1. Initial program 89.8%

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

      1. +-commutative 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-lft-out 89.8%

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

      4. associate-*l* 89.8%

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

      5. *-commutative 89.8%

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

      6. distribute-lft-out 89.8%

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

      7. associate-+r- 89.8%

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

      8. distribute-lft-out-- 89.8%

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

   3. Simplified 89.9%

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

      1. sub-neg 89.9%

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

      2. associate-*l* 89.9%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around inf 74.4%

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

      1. *-commutative 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*l* 74.4%

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

      4. *-commutative 74.4%

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

      5. unpow2 74.4%

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

   8. Simplified 74.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{+155}:\\ \;\;\;\;\left(x.im + x.im\right) + x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq -30 \lor \neg \left(x.im \leq 2.45 \cdot 10^{+31}\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 6: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -17 \lor \neg \left(x.im \leq 2.45 \cdot 10^{+31}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -17.0) (not (<= x.im 2.45e+31)))
   (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))
   (* x.im (* (* x.re x.re) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -17.0) || !(x_46_im <= 2.45e+31)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 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 <= (-17.0d0)) .or. (.not. (x_46im <= 2.45d+31))) then
        tmp = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    else
        tmp = x_46im * ((x_46re * x_46re) * 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 <= -17.0) || !(x_46_im <= 2.45e+31)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -17.0) or not (x_46_im <= 2.45e+31):
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -17.0) || !(x_46_im <= 2.45e+31))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -17.0) || ~((x_46_im <= 2.45e+31)))
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -17.0], N[Not[LessEqual[x$46$im, 2.45e+31]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -17 or 2.44999999999999998e31 < x.im

   1. Initial program 73.1%

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

      1. +-commutative 73.1%

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

      2. *-commutative 73.1%

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

      3. fma-def 76.4%

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

      4. *-commutative 76.4%

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

      5. distribute-lft-out 76.4%

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

      6. *-commutative 76.4%

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

   3. Simplified 76.4%

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

      1. fma-udef 73.1%

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

      2. distribute-lft-in 73.1%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      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-+ 83.5%

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

      17. *-commutative 83.5%

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

      18. difference-of-squares 98.9%

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

      19. associate-*r* 98.9%

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

      20. *-commutative 98.9%

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

   5. Applied egg-rr 98.9%

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

   if -17 < x.im < 2.44999999999999998e31

   1. Initial program 89.8%

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

      1. +-commutative 89.8%

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

      2. *-commutative 89.8%

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

      3. distribute-lft-out 89.8%

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

      4. associate-*l* 89.8%

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

      5. *-commutative 89.8%

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

      6. distribute-lft-out 89.8%

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

      7. associate-+r- 89.8%

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

      8. distribute-lft-out-- 89.8%

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

   3. Simplified 89.9%

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

      1. sub-neg 89.9%

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

      2. associate-*l* 89.9%

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

      3. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x.re around inf 74.4%

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

      1. *-commutative 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*l* 74.4%

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

      4. *-commutative 74.4%

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

      5. unpow2 74.4%

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

   8. Simplified 74.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -17 \lor \neg \left(x.im \leq 2.45 \cdot 10^{+31}\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 7: 53.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -8.5e+164)
   (+ (+ x.im x.im) (* x.im (* x.re x.im)))
   (* x.im (* (* x.re x.re) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -8.5e+164) {
		tmp = (x_46_im + x_46_im) + (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if (x_46im <= (-8.5d+164)) then
        tmp = (x_46im + x_46im) + (x_46im * (x_46re * x_46im))
    else
        tmp = x_46im * ((x_46re * x_46re) * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -8.5e+164) {
		tmp = (x_46_im + x_46_im) + (x_46_im * (x_46_re * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -8.5e+164:
		tmp = (x_46_im + x_46_im) + (x_46_im * (x_46_re * x_46_im))
	else:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -8.5e+164)
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_im * Float64(x_46_re * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -8.5e+164)
		tmp = (x_46_im + x_46_im) + (x_46_im * (x_46_re * x_46_im));
	else
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -8.5e+164], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$im * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -8.50000000000000027e164

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. *-commutative 58.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. distribute-lft-out 64.7%

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

      6. *-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. fma-udef 58.8%

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

      2. distribute-lft-in 58.8%

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

      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-+ 73.5%

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

      17. *-commutative 73.5%

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

      18. difference-of-squares 100.0%

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

      19. associate-*r* 100.0%

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

      20. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 47.8%

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

   7. Taylor expanded in x.re around 0 50.0%

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

      1. unpow2 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*r* 36.0%

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

   9. Simplified 36.0%

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

   if -8.50000000000000027e164 < x.im

   1. Initial program 85.3%

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

      1. +-commutative 85.3%

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

      2. *-commutative 85.3%

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

      3. distribute-lft-out 85.3%

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

      4. associate-*l* 85.3%

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

      5. *-commutative 85.3%

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

      6. distribute-lft-out 89.4%

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

      7. associate-+r- 89.4%

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

      8. distribute-lft-out-- 79.9%

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

   3. Simplified 80.0%

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

      1. sub-neg 80.0%

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

      2. associate-*l* 80.0%

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

      3. associate-*l* 85.9%

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

   5. Applied egg-rr 85.9%

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

   6. Taylor expanded in x.re around inf 57.9%

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

      1. *-commutative 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

      4. *-commutative 57.9%

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

      5. unpow2 57.9%

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

   8. Simplified 57.9%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\left(x.im + x.im\right) + x.im \cdot \left(x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 8: 54.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.2 \cdot 10^{+164}:\\ \;\;\;\;\left(x.im + x.im\right) + x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (<= x.im -2.2e+164)
   (+ (+ x.im x.im) (* x.re (* x.im x.im)))
   (* x.im (* (* x.re x.re) 3.0))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if (x_46_im <= -2.2e+164) {
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 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 <= (-2.2d+164)) then
        tmp = (x_46im + x_46im) + (x_46re * (x_46im * x_46im))
    else
        tmp = x_46im * ((x_46re * x_46re) * 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 <= -2.2e+164) {
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	} else {
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if x_46_im <= -2.2e+164:
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im))
	else:
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if (x_46_im <= -2.2e+164)
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(x_46_re * Float64(x_46_im * x_46_im)));
	else
		tmp = Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.2e+164)
		tmp = (x_46_im + x_46_im) + (x_46_re * (x_46_im * x_46_im));
	else
		tmp = x_46_im * ((x_46_re * x_46_re) * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[LessEqual[x$46$im, -2.2e+164], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(x$46$re * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.20000000000000006e164

   1. Initial program 58.8%

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

      1. +-commutative 58.8%

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

      2. *-commutative 58.8%

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

      3. fma-def 64.7%

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

      4. *-commutative 64.7%

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

      5. distribute-lft-out 64.7%

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

      6. *-commutative 64.7%

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

   3. Simplified 64.7%

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

      1. fma-udef 58.8%

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

      2. distribute-lft-in 58.8%

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

      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-+ 73.5%

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

      17. *-commutative 73.5%

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

      18. difference-of-squares 100.0%

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

      19. associate-*r* 100.0%

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

      20. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x.im around 0 47.8%

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

   7. Taylor expanded in x.re around 0 50.0%

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

      1. unpow2 50.0%

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

   9. Simplified 50.0%

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

   if -2.20000000000000006e164 < x.im

   1. Initial program 85.3%

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

      1. +-commutative 85.3%

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

      2. *-commutative 85.3%

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

      3. distribute-lft-out 85.3%

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

      4. associate-*l* 85.3%

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

      5. *-commutative 85.3%

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

      6. distribute-lft-out 89.4%

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

      7. associate-+r- 89.4%

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

      8. distribute-lft-out-- 79.9%

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

   3. Simplified 80.0%

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

      1. sub-neg 80.0%

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

      2. associate-*l* 80.0%

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

      3. associate-*l* 85.9%

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

   5. Applied egg-rr 85.9%

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

   6. Taylor expanded in x.re around inf 57.9%

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

      1. *-commutative 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-*l* 57.9%

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

      4. *-commutative 57.9%

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

      5. unpow2 57.9%

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

   8. Simplified 57.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.2 \cdot 10^{+164}:\\ \;\;\;\;\left(x.im + x.im\right) + x.re \cdot \left(x.im \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \left(\left(x.re \cdot x.re\right) \cdot 3\right)\\ \end{array} \]

Alternative 9: 49.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. *-commutative 81.8%

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

   3. distribute-lft-out 81.8%

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

   4. associate-*l* 81.8%

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

   5. *-commutative 81.8%

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

   6. distribute-lft-out 87.3%

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

   7. associate-+r- 87.3%

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

   8. distribute-lft-out-- 77.1%

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

3. Simplified 77.2%

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

   1. sub-neg 77.2%

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

   2. associate-*l* 77.2%

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

   3. associate-*l* 82.3%

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

5. Applied egg-rr 82.3%

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

6. Taylor expanded in x.re around inf 52.2%

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

   1. *-commutative 52.2%

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

   2. *-commutative 52.2%

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

   3. associate-*l* 52.2%

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

   4. *-commutative 52.2%

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

   5. unpow2 52.2%

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

8. Simplified 52.2%

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

9. Final simplification 52.2%

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

Alternative 10: 34.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative 81.8%

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

   2. *-commutative 81.8%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. associate-*r/ 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. distribute-lft-out-- 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. flip-+ 53.7%

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

3. Applied egg-rr 53.7%

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

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

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

   1. *-commutative 37.7%

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

   2. unpow2 37.7%

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

6. Simplified 37.7%

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

7. Final simplification 37.7%

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

Alternative 11: 3.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x.im + x.im \end{array} \]

FPCore

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

C

double code(double x_46_re, double x_46_im) {
	return 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
end function

Java

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

Python

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

Julia

function code(x_46_re, x_46_im)
	return 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;
end

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. *-commutative 81.8%

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

   3. fma-def 83.4%

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

   4. *-commutative 83.4%

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

   5. distribute-lft-out 83.4%

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

   6. *-commutative 83.4%

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

3. Simplified 83.4%

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

   1. fma-udef 81.8%

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

   2. distribute-lft-in 81.8%

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

   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-+ 53.7%

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

   17. *-commutative 53.7%

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

   18. difference-of-squares 61.1%

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

   19. associate-*r* 61.8%

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

   20. *-commutative 61.8%

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

5. Applied egg-rr 61.8%

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

6. Taylor expanded in x.im around 0 34.2%

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

7. Taylor expanded in x.re around 0 3.0%

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

   1. count-2 3.0%

      \[\leadsto \color{blue}{x.im + x.im} \]

9. Simplified 3.0%

   \[\leadsto \color{blue}{x.im + x.im} \]

10. Final simplification 3.0%

    \[\leadsto x.im + x.im \]

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_] := -3.0

Derivation

1. Initial program 81.8%

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

2. Taylor expanded in x.re around 0 55.6%

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

4. Simplified 2.6%

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

5. Final simplification 2.6%

   \[\leadsto -3 \]

Alternative 13: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return 2.6

Julia

function code(x_46_re, x_46_im)
	return 2.6
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. +-commutative 81.8%

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

   2. *-commutative 81.8%

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

   3. distribute-lft-out 81.8%

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

   4. associate-*l* 81.8%

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

   5. *-commutative 81.8%

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

   6. distribute-lft-out 87.3%

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

   7. associate-+r- 87.3%

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

   8. distribute-lft-out-- 77.1%

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

3. Simplified 77.2%

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

   1. flip3-- 12.8%

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

   2. div-inv 12.8%

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

   3. *-commutative 12.8%

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

   4. unpow-prod-down 12.8%

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

   5. metadata-eval 12.8%

      \[\leadsto \left(\color{blue}{27} \cdot {\left(\left(x.re \cdot x.re\right) \cdot x.im\right)}^{3} - {\left({x.im}^{3}\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)} \]

   6. associate-*l* 12.8%

      \[\leadsto \left(27 \cdot {\color{blue}{\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}}^{3} - {\left({x.im}^{3}\right)}^{3}\right) \cdot \frac{1}{\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)} \]

   7. pow-pow 12.8%

      \[\leadsto \left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - \color{blue}{{x.im}^{\left(3 \cdot 3\right)}}\right) \cdot \frac{1}{\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)} \]

   8. metadata-eval 12.8%

      \[\leadsto \left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{\color{blue}{9}}\right) \cdot \frac{1}{\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + \left({x.im}^{3} \cdot {x.im}^{3} + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}\right)} \]

   9. associate-+r+ 12.8%

      \[\leadsto \left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{9}\right) \cdot \frac{1}{\color{blue}{\left(\left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) + {x.im}^{3} \cdot {x.im}^{3}\right) + \left(\left(\left(x.re \cdot x.re\right) \cdot x.im\right) \cdot 3\right) \cdot {x.im}^{3}}} \]

5. Applied egg-rr 10.0%

   \[\leadsto \color{blue}{\left(27 \cdot {\left(x.re \cdot \left(x.re \cdot x.im\right)\right)}^{3} - {x.im}^{9}\right) \cdot \frac{1}{\left(9 \cdot \left(\left(x.im \cdot x.im\right) \cdot {x.re}^{4}\right) + {x.im}^{6}\right) + \left(x.re \cdot x.re\right) \cdot \left(\left(x.im \cdot 3\right) \cdot {x.im}^{3}\right)}} \]

7. Simplified 2.8%

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

8. Final simplification 2.8%

   \[\leadsto 2.6 \]

Developer target: 91.8% 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 2023274 
(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)))