math.cube on complex, imaginary part

Percentage Accurate: 83.0% → 99.8%

Time: 11.5s

Alternatives: 13

Speedup: 1.1×

• 43.4% 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 91.5%

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

      1. *-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. difference-of-squares 91.5%

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

      4. associate-*l* 99.8%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.8%

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

   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 19.4%

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

      4. *-commutative 19.4%

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

      5. distribute-lft-out 19.4%

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

      6. *-commutative 19.4%

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

   3. Simplified 19.4%

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

      1. fma-udef 0.0%

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

      2. distribute-lft-in 0.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 33.3%

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

      9. distribute-lft-in 33.3%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 38.9%

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

      16. *-commutative 38.9%

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

      17. difference-of-squares 100.0%

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

      18. associate-*r* 100.0%

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

      19. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

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

Alternative 2: 97.4% accurate, 0.1× speedup

Math

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

FPCore

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

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - Math.pow(x_46_im, 3.0);
	} 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):
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - math.pow(x_46_im, 3.0)
	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)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(Float64(x_46_re * Float64(x_46_im * Float64(x_46_re * 3.0))) - (x_46_im ^ 3.0));
	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)
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = (x_46_re * (x_46_im * (x_46_re * 3.0))) - (x_46_im ^ 3.0);
	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_] := If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re * N[(x$46$im * N[(x$46$re * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $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]]

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 91.5%

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

      1. +-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. distribute-lft-out 91.5%

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

      4. associate-*l* 91.5%

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

      5. *-commutative 91.5%

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

      6. distribute-lft-out 91.4%

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

      7. associate-+r- 91.4%

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

      8. distribute-lft-out-- 90.1%

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

   3. Simplified 90.2%

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

      1. sub-neg 90.2%

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

      2. associate-*l* 90.2%

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

      3. associate-*l* 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x.re around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

   8. Simplified 98.4%

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

      1. unsub-neg 98.4%

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

      2. *-commutative 98.4%

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

   10. Applied egg-rr 98.4%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      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 19.4%

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

      4. *-commutative 19.4%

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

      5. distribute-lft-out 19.4%

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

      6. *-commutative 19.4%

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

   3. Simplified 19.4%

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

      1. fma-udef 0.0%

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

      2. distribute-lft-in 0.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 33.3%

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

      9. distribute-lft-in 33.3%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 38.9%

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

      16. *-commutative 38.9%

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

      17. difference-of-squares 100.0%

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

      18. associate-*r* 100.0%

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

      19. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 98.6%

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

Alternative 3: 97.4% accurate, 0.1× speedup

Math

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

FPCore

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

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - Math.pow(x_46_im, 3.0);
	} else {
		tmp = (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):
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= math.inf:
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - math.pow(x_46_im, 3.0)
	else:
		tmp = (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)
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))) <= Inf)
		tmp = Float64(Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0))) - (x_46_im ^ 3.0));
	else
		tmp = Float64(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)
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))) <= Inf)
		tmp = (x_46_re * (x_46_re * (x_46_im * 3.0))) - (x_46_im ^ 3.0);
	else
		tmp = (x_46_im + 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_] := If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x$46$im, 3.0], $MachinePrecision]), $MachinePrecision], N[(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)) < +inf.0

   1. Initial program 91.5%

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

      1. +-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. distribute-lft-out 91.5%

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

      4. associate-*l* 91.5%

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

      5. *-commutative 91.5%

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

      6. distribute-lft-out 91.4%

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

      7. associate-+r- 91.4%

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

      8. distribute-lft-out-- 90.1%

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

   3. Simplified 90.2%

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

      1. sub-neg 90.2%

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

      2. associate-*l* 90.2%

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

      3. associate-*l* 98.5%

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

   5. Applied egg-rr 98.5%

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

   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 19.4%

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

      4. *-commutative 19.4%

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

      5. distribute-lft-out 19.4%

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

      6. *-commutative 19.4%

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

   3. Simplified 19.4%

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

      1. fma-udef 0.0%

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

      2. distribute-lft-in 0.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 33.3%

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

      9. distribute-lft-in 33.3%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 38.9%

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

      16. *-commutative 38.9%

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

      17. difference-of-squares 100.0%

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

      18. associate-*r* 100.0%

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

      19. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 98.7%

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

Alternative 4: 94.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re - x.im \cdot x.im\\ t_1 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;x.im \cdot t_0 + t_1 \leq 2 \cdot 10^{+286}:\\ \;\;\;\;t_1 + \frac{x.im}{\frac{1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (- (* x.re x.re) (* x.im x.im)))
        (t_1 (* x.re (+ (* x.re x.im) (* x.re x.im)))))
   (if (<= (+ (* x.im t_0) t_1) 2e+286)
     (+ t_1 (/ x.im (/ 1.0 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_re * x_46_re) - (x_46_im * 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 * t_0) + t_1) <= 2e+286) {
		tmp = t_1 + (x_46_im / (1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = (x_46re * x_46re) - (x_46im * x_46im)
    t_1 = x_46re * ((x_46re * x_46im) + (x_46re * x_46im))
    if (((x_46im * t_0) + t_1) <= 2d+286) then
        tmp = t_1 + (x_46im / (1.0d0 / 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_re * x_46_re) - (x_46_im * 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 * t_0) + t_1) <= 2e+286) {
		tmp = t_1 + (x_46_im / (1.0 / 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_re * x_46_re) - (x_46_im * 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 * t_0) + t_1) <= 2e+286:
		tmp = t_1 + (x_46_im / (1.0 / 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_re * x_46_re) - Float64(x_46_im * 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 * t_0) + t_1) <= 2e+286)
		tmp = Float64(t_1 + Float64(x_46_im / Float64(1.0 / 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_re * x_46_re) - (x_46_im * 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 * t_0) + t_1) <= 2e+286)
		tmp = t_1 + (x_46_im / (1.0 / 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$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $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 * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], 2e+286], N[(t$95$1 + N[(x$46$im / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $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]]]]

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)) < 2.00000000000000007e286

   1. Initial program 92.4%

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

      1. difference-of-squares 92.4%

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

      2. flip-+ 92.4%

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

      3. associate-*l/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. *-commutative 82.5%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

      4. *-un-lft-identity 82.5%

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

      5. times-frac 92.4%

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

      6. *-inverses 92.4%

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

   5. Applied egg-rr 92.4%

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

   if 2.00000000000000007e286 < (+.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 49.9%

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

      1. +-commutative 49.9%

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

      2. *-commutative 49.9%

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

      3. fma-def 58.4%

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

      4. *-commutative 58.4%

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

      5. distribute-lft-out 58.4%

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

      6. *-commutative 58.4%

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

   3. Simplified 58.4%

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

      1. fma-udef 49.9%

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

      2. distribute-lft-in 49.9%

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

      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%

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

      8. flip-+ 64.4%

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

      9. distribute-lft-in 64.4%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 66.8%

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

      16. *-commutative 66.8%

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

      17. difference-of-squares 93.3%

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

      18. associate-*r* 94.1%

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

      19. *-commutative 94.1%

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

   5. Applied egg-rr 94.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 2 \cdot 10^{+286}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \frac{x.im}{\frac{1}{x.re \cdot x.re - x.im \cdot x.im}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 5: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{if}\;t_0 \leq 10^{+262}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \end{array} \]

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 1e+262)
     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 <= 1e+262) {
		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 <= 1d+262) 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 <= 1e+262) {
		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 <= 1e+262:
		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 <= 1e+262)
		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 <= 1e+262)
		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, 1e+262], 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)) < 1e262

   1. Initial program 92.3%

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

   if 1e262 < (+.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 50.5%

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

      1. +-commutative 50.5%

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

      2. *-commutative 50.5%

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

      3. fma-def 58.8%

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

      4. *-commutative 58.8%

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

      5. distribute-lft-out 58.8%

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

      6. *-commutative 58.8%

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

   3. Simplified 58.8%

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

      1. fma-udef 50.5%

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

      2. distribute-lft-in 50.5%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 64.8%

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

      9. distribute-lft-in 64.8%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 67.2%

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

      16. *-commutative 67.2%

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

      17. difference-of-squares 93.4%

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

      18. associate-*r* 94.1%

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

      19. *-commutative 94.1%

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

   5. Applied egg-rr 94.1%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) \leq 10^{+262}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 6: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ t_2 := x.re \cdot x.re - x.im \cdot x.im\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -2.8 \cdot 10^{-104}:\\ \;\;\;\;t_0 + \frac{x.im}{\frac{1}{t_2}}\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{-110}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+41}:\\ \;\;\;\;x.im \cdot t_2 + t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im) (* x.re x.im))))
        (t_1 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im)))))
        (t_2 (- (* x.re x.re) (* x.im x.im))))
   (if (<= x.im -1e+139)
     t_1
     (if (<= x.im -2.8e-104)
       (+ t_0 (/ x.im (/ 1.0 t_2)))
       (if (<= x.im 4e-110)
         (* 3.0 (* x.re (* x.re x.im)))
         (if (<= x.im 4e+41) (+ (* x.im t_2) t_0) t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double t_2 = (x_46_re * x_46_re) - (x_46_im * x_46_im);
	double tmp;
	if (x_46_im <= -1e+139) {
		tmp = t_1;
	} else if (x_46_im <= -2.8e-104) {
		tmp = t_0 + (x_46_im / (1.0 / t_2));
	} else if (x_46_im <= 4e-110) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 4e+41) {
		tmp = (x_46_im * t_2) + 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) :: t_2
    real(8) :: tmp
    t_0 = x_46re * ((x_46re * x_46im) + (x_46re * x_46im))
    t_1 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    t_2 = (x_46re * x_46re) - (x_46im * x_46im)
    if (x_46im <= (-1d+139)) then
        tmp = t_1
    else if (x_46im <= (-2.8d-104)) then
        tmp = t_0 + (x_46im / (1.0d0 / t_2))
    else if (x_46im <= 4d-110) then
        tmp = 3.0d0 * (x_46re * (x_46re * x_46im))
    else if (x_46im <= 4d+41) then
        tmp = (x_46im * t_2) + 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_re * ((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 t_2 = (x_46_re * x_46_re) - (x_46_im * x_46_im);
	double tmp;
	if (x_46_im <= -1e+139) {
		tmp = t_1;
	} else if (x_46_im <= -2.8e-104) {
		tmp = t_0 + (x_46_im / (1.0 / t_2));
	} else if (x_46_im <= 4e-110) {
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	} else if (x_46_im <= 4e+41) {
		tmp = (x_46_im * t_2) + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	t_2 = (x_46_re * x_46_re) - (x_46_im * x_46_im)
	tmp = 0
	if x_46_im <= -1e+139:
		tmp = t_1
	elif x_46_im <= -2.8e-104:
		tmp = t_0 + (x_46_im / (1.0 / t_2))
	elif x_46_im <= 4e-110:
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im))
	elif x_46_im <= 4e+41:
		tmp = (x_46_im * t_2) + t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_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))))
	t_2 = Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (x_46_im <= -1e+139)
		tmp = t_1;
	elseif (x_46_im <= -2.8e-104)
		tmp = Float64(t_0 + Float64(x_46_im / Float64(1.0 / t_2)));
	elseif (x_46_im <= 4e-110)
		tmp = Float64(3.0 * Float64(x_46_re * Float64(x_46_re * x_46_im)));
	elseif (x_46_im <= 4e+41)
		tmp = Float64(Float64(x_46_im * t_2) + 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_re * ((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)));
	t_2 = (x_46_re * x_46_re) - (x_46_im * x_46_im);
	tmp = 0.0;
	if (x_46_im <= -1e+139)
		tmp = t_1;
	elseif (x_46_im <= -2.8e-104)
		tmp = t_0 + (x_46_im / (1.0 / t_2));
	elseif (x_46_im <= 4e-110)
		tmp = 3.0 * (x_46_re * (x_46_re * x_46_im));
	elseif (x_46_im <= 4e+41)
		tmp = (x_46_im * t_2) + t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$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]}, Block[{t$95$2 = N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1e+139], t$95$1, If[LessEqual[x$46$im, -2.8e-104], N[(t$95$0 + N[(x$46$im / N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4e-110], N[(3.0 * N[(x$46$re * N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4e+41], N[(N[(x$46$im * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -1.00000000000000003e139 or 4.00000000000000002e41 < x.im

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

      2. *-commutative 59.1%

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

      3. fma-def 67.0%

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

      4. *-commutative 67.0%

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

      5. distribute-lft-out 67.0%

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

      6. *-commutative 67.0%

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

   3. Simplified 67.0%

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

      1. fma-udef 59.1%

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

      2. distribute-lft-in 59.1%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 72.7%

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

      9. distribute-lft-in 72.7%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 75.0%

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

      16. *-commutative 75.0%

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

      17. difference-of-squares 100.0%

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

      18. associate-*r* 100.0%

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

      19. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -1.00000000000000003e139 < x.im < -2.8e-104

   1. Initial program 93.2%

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

      1. difference-of-squares 93.2%

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

      2. flip-+ 93.2%

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

      3. associate-*l/ 86.6%

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

   3. Applied egg-rr 86.6%

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

      1. *-commutative 86.6%

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

      2. clear-num 86.4%

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

      3. un-div-inv 86.5%

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

      4. *-un-lft-identity 86.5%

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

      5. times-frac 93.3%

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

      6. *-inverses 93.3%

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

   5. Applied egg-rr 93.3%

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

   if -2.8e-104 < x.im < 4.0000000000000002e-110

   1. Initial program 82.0%

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

      1. +-commutative 82.0%

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

      2. *-commutative 82.0%

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

      3. distribute-lft-out 82.0%

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

      4. associate-*l* 82.0%

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

      5. *-commutative 82.0%

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

      6. distribute-lft-out 82.0%

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

      7. associate-+r- 82.0%

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

      8. distribute-lft-out-- 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x.re around inf 82.0%

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

      1. pow1 82.0%

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

      2. pow2 82.0%

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

      3. associate-*l* 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 4.0000000000000002e-110 < x.im < 4.00000000000000002e41

   1. Initial program 97.2%

      \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -2.8 \cdot 10^{-104}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \frac{x.im}{\frac{1}{x.re \cdot x.re - x.im \cdot x.im}}\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{-110}:\\ \;\;\;\;3 \cdot \left(x.re \cdot \left(x.re \cdot x.im\right)\right)\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{+41}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 7: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\frac{1}{x.re \cdot x.re - x.im \cdot x.im}} + x.re \cdot 0\\ t_1 := \left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -4.3 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 3.5 \cdot 10^{-103}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 10:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im (/ 1.0 (- (* x.re x.re) (* x.im x.im)))) (* x.re 0.0)))
        (t_1 (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))))
   (if (<= x.im -6.2e+173)
     t_1
     (if (<= x.im -4.3e-101)
       t_0
       (if (<= x.im 3.5e-103)
         (+ (* x.re (+ (* x.re x.im) (* x.re x.im))) (* (* x.re x.re) x.im))
         (if (<= x.im 10.0) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im / (1.0 / ((x_46_re * x_46_re) - (x_46_im * x_46_im)))) + (x_46_re * 0.0);
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -6.2e+173) {
		tmp = t_1;
	} else if (x_46_im <= -4.3e-101) {
		tmp = t_0;
	} else if (x_46_im <= 3.5e-103) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	} else if (x_46_im <= 10.0) {
		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 / (1.0d0 / ((x_46re * x_46re) - (x_46im * x_46im)))) + (x_46re * 0.0d0)
    t_1 = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    if (x_46im <= (-6.2d+173)) then
        tmp = t_1
    else if (x_46im <= (-4.3d-101)) then
        tmp = t_0
    else if (x_46im <= 3.5d-103) then
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + ((x_46re * x_46re) * x_46im)
    else if (x_46im <= 10.0d0) 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 / (1.0 / ((x_46_re * x_46_re) - (x_46_im * x_46_im)))) + (x_46_re * 0.0);
	double t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	double tmp;
	if (x_46_im <= -6.2e+173) {
		tmp = t_1;
	} else if (x_46_im <= -4.3e-101) {
		tmp = t_0;
	} else if (x_46_im <= 3.5e-103) {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	} else if (x_46_im <= 10.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im / (1.0 / ((x_46_re * x_46_re) - (x_46_im * x_46_im)))) + (x_46_re * 0.0)
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	tmp = 0
	if x_46_im <= -6.2e+173:
		tmp = t_1
	elif x_46_im <= -4.3e-101:
		tmp = t_0
	elif x_46_im <= 3.5e-103:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im)
	elif x_46_im <= 10.0:
		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(1.0 / Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im)))) + Float64(x_46_re * 0.0))
	t_1 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))))
	tmp = 0.0
	if (x_46_im <= -6.2e+173)
		tmp = t_1;
	elseif (x_46_im <= -4.3e-101)
		tmp = t_0;
	elseif (x_46_im <= 3.5e-103)
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(Float64(x_46_re * x_46_re) * x_46_im));
	elseif (x_46_im <= 10.0)
		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 / (1.0 / ((x_46_re * x_46_re) - (x_46_im * x_46_im)))) + (x_46_re * 0.0);
	t_1 = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	tmp = 0.0;
	if (x_46_im <= -6.2e+173)
		tmp = t_1;
	elseif (x_46_im <= -4.3e-101)
		tmp = t_0;
	elseif (x_46_im <= 3.5e-103)
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	elseif (x_46_im <= 10.0)
		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[(1.0 / N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * 0.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -6.2e+173], t$95$1, If[LessEqual[x$46$im, -4.3e-101], t$95$0, If[LessEqual[x$46$im, 3.5e-103], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 10.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -6.2e173 or 10 < x.im

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. *-commutative 59.0%

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

      3. fma-def 65.0%

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

      4. *-commutative 65.0%

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

      5. distribute-lft-out 65.0%

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

      6. *-commutative 65.0%

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

   3. Simplified 65.0%

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

      1. fma-udef 59.0%

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

      2. distribute-lft-in 59.0%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 70.1%

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

      9. distribute-lft-in 70.1%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 72.5%

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

      16. *-commutative 72.5%

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

      17. difference-of-squares 99.0%

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

      18. associate-*r* 99.0%

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

      19. *-commutative 99.0%

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

   5. Applied egg-rr 99.0%

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

   if -6.2e173 < x.im < -4.2999999999999997e-101 or 3.50000000000000016e-103 < x.im < 10

   1. Initial program 92.7%

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

      1. difference-of-squares 92.7%

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

      2. flip-+ 92.7%

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

      3. associate-*l/ 84.7%

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

   3. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. clear-num 84.5%

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

      3. un-div-inv 84.6%

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

      4. *-un-lft-identity 84.6%

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

      5. times-frac 92.8%

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

      6. *-inverses 92.8%

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

   5. Applied egg-rr 92.8%

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

      1. *-commutative 92.8%

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

      2. rem-cbrt-cube 79.7%

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

      3. add-sqr-sqrt 29.4%

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

      4. sqrt-unprod 68.3%

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

      5. sqr-neg 68.3%

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

      6. sqrt-unprod 46.8%

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

      7. add-sqr-sqrt 70.0%

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

      8. neg-mul-1 70.0%

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

      9. cbrt-prod 70.0%

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

      10. metadata-eval 70.0%

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

      11. metadata-eval 70.0%

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

      12. add-cbrt-cube 70.0%

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

      13. rem-cbrt-cube 74.4%

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

      14. neg-mul-1 74.4%

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

      15. cancel-sign-sub-inv 74.4%

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

      16. +-inverses 79.8%

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

   7. Applied egg-rr 79.8%

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

   if -4.2999999999999997e-101 < x.im < 3.50000000000000016e-103

   1. Initial program 82.7%

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

      1. difference-of-squares 82.7%

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

      2. flip-+ 82.7%

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

      3. associate-*l/ 70.8%

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

   3. Applied egg-rr 70.8%

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

   4. Taylor expanded in x.re around inf 82.7%

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

   6. Simplified 82.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.2 \cdot 10^{+173}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{elif}\;x.im \leq -4.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.im}{\frac{1}{x.re \cdot x.re - x.im \cdot x.im}} + x.re \cdot 0\\ \mathbf{elif}\;x.im \leq 3.5 \cdot 10^{-103}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot x.im\\ \mathbf{elif}\;x.im \leq 10:\\ \;\;\;\;\frac{x.im}{\frac{1}{x.re \cdot x.re - x.im \cdot x.im}} + x.re \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \end{array} \]

Alternative 8: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -49000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -49000000.0) (not (<= x.im 0.001)))
   (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))
   (+ (* x.re (+ (* x.re x.im) (* x.re x.im))) (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -49000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-49000000.0d0)) .or. (.not. (x_46im <= 0.001d0))) then
        tmp = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    else
        tmp = (x_46re * ((x_46re * x_46im) + (x_46re * x_46im))) + ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -49000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -49000000.0) or not (x_46_im <= 0.001):
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -49000000.0) || !(x_46_im <= 0.001))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_re + x_46_im) * Float64(x_46_im * Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im))) + Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -49000000.0) || ~((x_46_im <= 0.001)))
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))) + ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -49000000.0], N[Not[LessEqual[x$46$im, 0.001]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $MachinePrecision] + N[(N[(x$46$re + x$46$im), $MachinePrecision] * N[(x$46$im * N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.9e7 or 1e-3 < x.im

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. *-commutative 69.9%

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

      3. fma-def 75.8%

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

      4. *-commutative 75.8%

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

      5. distribute-lft-out 75.8%

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

      6. *-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. fma-udef 69.9%

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

      2. distribute-lft-in 69.9%

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

      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%

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

      8. flip-+ 75.4%

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

      9. distribute-lft-in 75.4%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 78.9%

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

      16. *-commutative 78.9%

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

      17. difference-of-squares 97.2%

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

      18. associate-*r* 97.2%

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

      19. *-commutative 97.2%

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

   5. Applied egg-rr 97.2%

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

   if -4.9e7 < x.im < 1e-3

   1. Initial program 86.3%

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

      1. difference-of-squares 86.3%

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

      2. flip-+ 86.3%

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

      3. associate-*l/ 75.1%

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

   3. Applied egg-rr 75.1%

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

   4. Taylor expanded in x.re around inf 70.3%

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

   6. Simplified 70.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -49000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \left(x.re \cdot x.re\right) \cdot x.im\\ \end{array} \]

Alternative 9: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -50000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -50000000.0) (not (<= x.im 0.001)))
   (+ (* x.im (- (* x.re x.re) (* x.im x.im))) (+ x.im x.im))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -50000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-50000000.0d0)) .or. (.not. (x_46im <= 0.001d0))) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46im + x_46im)
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -50000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -50000000.0) or not (x_46_im <= 0.001):
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im)
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -50000000.0) || !(x_46_im <= 0.001))
		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(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -50000000.0) || ~((x_46_im <= 0.001)))
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -50000000.0], N[Not[LessEqual[x$46$im, 0.001]], $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[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5e7 or 1e-3 < x.im

   1. Initial program 69.9%

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

      1. *-commutative 69.9%

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

      2. flip-+ 0.0%

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

      3. +-inverses 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. flip-+ 75.4%

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

      8. *-commutative 75.4%

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

      9. distribute-lft-in 75.4%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 78.9%

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

   3. Applied egg-rr 78.9%

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

   if -5e7 < x.im < 1e-3

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

      2. *-commutative 86.3%

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

      3. distribute-lft-out 86.3%

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

      4. associate-*l* 86.3%

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

      5. *-commutative 86.3%

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

      6. distribute-lft-out 86.2%

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

      7. associate-+r- 86.2%

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

      8. distribute-lft-out-- 86.2%

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

   3. Simplified 86.3%

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

      1. sub-neg 86.3%

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

      2. associate-*l* 86.3%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around inf 70.2%

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

      1. *-commutative 70.2%

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

      2. unpow2 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -50000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 10: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -59000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -59000000.0) (not (<= x.im 0.001)))
   (+ (+ x.im x.im) (* (+ x.re x.im) (* x.im (- x.re x.im))))
   (* 3.0 (* (* x.re x.re) x.im))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -59000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-59000000.0d0)) .or. (.not. (x_46im <= 0.001d0))) then
        tmp = (x_46im + x_46im) + ((x_46re + x_46im) * (x_46im * (x_46re - x_46im)))
    else
        tmp = 3.0d0 * ((x_46re * x_46re) * x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -59000000.0) || !(x_46_im <= 0.001)) {
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	} else {
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -59000000.0) or not (x_46_im <= 0.001):
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)))
	else:
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -59000000.0) || !(x_46_im <= 0.001))
		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(3.0 * Float64(Float64(x_46_re * x_46_re) * x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -59000000.0) || ~((x_46_im <= 0.001)))
		tmp = (x_46_im + x_46_im) + ((x_46_re + x_46_im) * (x_46_im * (x_46_re - x_46_im)));
	else
		tmp = 3.0 * ((x_46_re * x_46_re) * x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -59000000.0], N[Not[LessEqual[x$46$im, 0.001]], $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[(3.0 * N[(N[(x$46$re * x$46$re), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.9e7 or 1e-3 < x.im

   1. Initial program 69.9%

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

      1. +-commutative 69.9%

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

      2. *-commutative 69.9%

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

      3. fma-def 75.8%

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

      4. *-commutative 75.8%

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

      5. distribute-lft-out 75.8%

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

      6. *-commutative 75.8%

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

   3. Simplified 75.8%

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

      1. fma-udef 69.9%

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

      2. distribute-lft-in 69.9%

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

      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%

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

      8. flip-+ 75.4%

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

      9. distribute-lft-in 75.4%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 78.9%

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

      16. *-commutative 78.9%

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

      17. difference-of-squares 97.2%

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

      18. associate-*r* 97.2%

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

      19. *-commutative 97.2%

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

   5. Applied egg-rr 97.2%

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

   if -5.9e7 < x.im < 1e-3

   1. Initial program 86.3%

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

      1. +-commutative 86.3%

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

      2. *-commutative 86.3%

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

      3. distribute-lft-out 86.3%

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

      4. associate-*l* 86.3%

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

      5. *-commutative 86.3%

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

      6. distribute-lft-out 86.2%

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

      7. associate-+r- 86.2%

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

      8. distribute-lft-out-- 86.2%

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

   3. Simplified 86.3%

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

      1. sub-neg 86.3%

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

      2. associate-*l* 86.3%

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

      3. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x.re around inf 70.2%

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

      1. *-commutative 70.2%

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

      2. unpow2 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -59000000 \lor \neg \left(x.im \leq 0.001\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.re + x.im\right) \cdot \left(x.im \cdot \left(x.re - x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\left(x.re \cdot x.re\right) \cdot x.im\right)\\ \end{array} \]

Alternative 11: 50.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

   1. +-commutative 78.6%

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

   2. *-commutative 78.6%

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

   3. distribute-lft-out 78.6%

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

   4. associate-*l* 78.6%

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

   5. *-commutative 78.6%

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

   6. distribute-lft-out 84.0%

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

   7. associate-+r- 84.0%

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

   8. distribute-lft-out-- 77.4%

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

3. Simplified 77.5%

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

   1. sub-neg 77.5%

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

   2. associate-*l* 77.5%

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

   3. associate-*l* 84.6%

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

5. Applied egg-rr 84.6%

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

6. Taylor expanded in x.re around inf 48.5%

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

   1. *-commutative 48.5%

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

   2. unpow2 48.5%

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

8. Simplified 48.5%

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

9. Final simplification 48.5%

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

Alternative 12: 34.4% 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 78.6%

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

   1. *-commutative 78.6%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. flip-+ 61.6%

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

   8. *-commutative 61.6%

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

   9. distribute-rgt-in 61.6%

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

   10. *-commutative 61.6%

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

   11. flip-+ 0.0%

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

   12. clear-num 0.0%

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

   13. *-commutative 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. *-commutative 0.0%

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

   17. *-commutative 0.0%

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

   18. +-inverses 0.0%

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

   19. +-inverses 0.0%

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

   20. flip-+ 46.8%

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

3. Applied egg-rr 46.8%

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

4. Taylor expanded in x.re around inf 33.2%

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

6. Simplified 33.2%

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

7. Final simplification 33.2%

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

Alternative 13: 2.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x_46_re, x_46_im):
	return -3.0

Julia

function code(x_46_re, x_46_im)
	return -3.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

2. Taylor expanded in x.re around 0 57.9%

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

4. Simplified 2.7%

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

5. Final simplification 2.7%

   \[\leadsto -3 \]

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 2023279 
(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)))