math.cube on complex, imaginary part

Percentage Accurate: 83.2% → 99.7%

Time: 11.2s

Alternatives: 12

Speedup: 1.1×

• 43.7% 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: 83.2% 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: 99.7% accurate, 0.1× speedup

Math

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

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) (* t_0 (+ 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_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) + (t_0 * (x_46_re + x_46_im));
	}
	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(t_0 * Float64(x_46_re + x_46_im)));
	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[(t$95$0 * N[(x$46$re + x$46$im), $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 91.6%

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

      1. *-commutative 91.6%

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

      2. *-commutative 91.6%

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

      3. difference-of-squares 91.6%

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

      4. associate-*l* 99.7%

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

      5. fma-def 99.8%

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

      6. *-commutative 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 26.3%

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

      4. *-commutative 26.3%

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

      5. distribute-lft-out 26.3%

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

      6. *-commutative 26.3%

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

   3. Simplified 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im + x.im\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   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-+ 26.3%

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

      9. distribute-lft-in 26.3%

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

      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-+ 26.3%

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

      16. *-commutative 26.3%

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

      17. difference-of-squares 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%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im) (* x.re x.im)))))
   (if (<= (+ (* x.im (- (* x.re x.re) (* x.im x.im))) t_0) INFINITY)
     (+ t_0 (/ (+ x.re x.im) (/ (/ 1.0 x.im) (- x.re x.im))))
     (+ (+ x.im x.im) (* (* x.im (- x.re 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_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= ((double) INFINITY)) {
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - 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 t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double tmp;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= Double.POSITIVE_INFINITY) {
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / (x_46_re - x_46_im)));
	} else {
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - 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_re * x_46_im))
	tmp = 0
	if ((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= math.inf:
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / (x_46_re - x_46_im)))
	else:
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))
	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)))
	tmp = 0.0
	if (Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = Float64(t_0 + Float64(Float64(x_46_re + x_46_im) / Float64(Float64(1.0 / x_46_im) / Float64(x_46_re - x_46_im))));
	else
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im * Float64(x_46_re - 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_re * x_46_im));
	tmp = 0.0;
	if (((x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0) <= Inf)
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / (x_46_re - x_46_im)));
	else
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - 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[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], Infinity], N[(t$95$0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] / N[(N[(1.0 / x$46$im), $MachinePrecision] / N[(x$46$re - x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + x$46$im), $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]]]

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 91.6%

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

      1. difference-of-squares 91.6%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

      4. flip-+ 91.6%

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

      5. associate-*l/ 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. expm1-log1p-u 58.3%

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

      2. expm1-udef 49.4%

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

      3. associate-/l* 50.7%

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

   5. Applied egg-rr 50.7%

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

      1. expm1-def 62.0%

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

      2. expm1-log1p 91.5%

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

      3. difference-of-squares 91.5%

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

      4. associate-/l* 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

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

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 26.3%

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

      4. *-commutative 26.3%

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

      5. distribute-lft-out 26.3%

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

      6. *-commutative 26.3%

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

   3. Simplified 26.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, x.re \cdot \left(x.im + x.im\right), x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right)\right)} \]
   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-+ 26.3%

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

      9. distribute-lft-in 26.3%

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

      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-+ 26.3%

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

      16. *-commutative 26.3%

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

      17. difference-of-squares 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%

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

Alternative 3: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := \left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -4.5 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 1.18 \cdot 10^{-125}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0
         (+
          (* x.im (- (* x.re x.re) (* x.im x.im)))
          (* x.re (+ (* x.re x.im) (* x.re x.im)))))
        (t_1 (+ (+ x.im x.im) (* (* x.im (- x.re x.im)) (+ x.re x.im)))))
   (if (<= x.im -1e+152)
     t_1
     (if (<= x.im -4.5e-71)
       t_0
       (if (<= x.im 1.18e-125)
         (* x.re (* x.re (* x.im 3.0)))
         (if (<= x.im 1e+82) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	double tmp;
	if (x_46_im <= -1e+152) {
		tmp = t_1;
	} else if (x_46_im <= -4.5e-71) {
		tmp = t_0;
	} else if (x_46_im <= 1.18e-125) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_im <= 1e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46re * ((x_46re * x_46im) + (x_46re * x_46im)))
    t_1 = (x_46im + x_46im) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
    if (x_46im <= (-1d+152)) then
        tmp = t_1
    else if (x_46im <= (-4.5d-71)) then
        tmp = t_0
    else if (x_46im <= 1.18d-125) then
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    else if (x_46im <= 1d+82) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	double t_1 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	double tmp;
	if (x_46_im <= -1e+152) {
		tmp = t_1;
	} else if (x_46_im <= -4.5e-71) {
		tmp = t_0;
	} else if (x_46_im <= 1.18e-125) {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	} else if (x_46_im <= 1e+82) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)))
	t_1 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))
	tmp = 0
	if x_46_im <= -1e+152:
		tmp = t_1
	elif x_46_im <= -4.5e-71:
		tmp = t_0
	elif x_46_im <= 1.18e-125:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	elif x_46_im <= 1e+82:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(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_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
	tmp = 0.0
	if (x_46_im <= -1e+152)
		tmp = t_1;
	elseif (x_46_im <= -4.5e-71)
		tmp = t_0;
	elseif (x_46_im <= 1.18e-125)
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	elseif (x_46_im <= 1e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im)));
	t_1 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	tmp = 0.0;
	if (x_46_im <= -1e+152)
		tmp = t_1;
	elseif (x_46_im <= -4.5e-71)
		tmp = t_0;
	elseif (x_46_im <= 1.18e-125)
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	elseif (x_46_im <= 1e+82)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + x$46$im), $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]}, If[LessEqual[x$46$im, -1e+152], t$95$1, If[LessEqual[x$46$im, -4.5e-71], t$95$0, If[LessEqual[x$46$im, 1.18e-125], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1e+82], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1e152 or 9.9999999999999996e81 < x.im

   1. Initial program 73.6%

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

      1. +-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. fma-def 80.5%

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

      4. *-commutative 80.5%

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

      5. distribute-lft-out 80.5%

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

      6. *-commutative 80.5%

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

   3. Simplified 80.5%

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

      1. fma-udef 73.6%

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

      2. distribute-lft-in 73.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 72.2%

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

      9. distribute-lft-in 72.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 80.5%

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

      16. *-commutative 80.5%

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

      17. difference-of-squares 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 -1e152 < x.im < -4.5000000000000002e-71 or 1.17999999999999994e-125 < x.im < 9.9999999999999996e81

   1. Initial program 99.7%

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

   if -4.5000000000000002e-71 < x.im < 1.17999999999999994e-125

   1. Initial program 79.4%

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

      1. difference-of-squares 79.4%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. flip-+ 79.3%

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

      5. associate-*l/ 74.7%

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

   3. Applied egg-rr 74.7%

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

      1. expm1-log1p-u 62.2%

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

      2. expm1-udef 49.6%

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

      3. associate-/l* 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. expm1-def 64.9%

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

      2. expm1-log1p 79.4%

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

      3. difference-of-squares 79.4%

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

      4. associate-/l* 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x.im around 0 77.6%

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

      1. distribute-lft1-in 77.6%

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

      2. metadata-eval 77.6%

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

      3. associate-*r* 77.5%

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

      4. *-commutative 77.5%

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

      5. metadata-eval 77.5%

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

      6. distribute-rgt1-in 77.5%

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

      7. unpow2 77.5%

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

      8. associate-*r* 97.8%

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

      9. distribute-rgt1-in 97.8%

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

      10. metadata-eval 97.8%

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

      11. *-commutative 97.8%

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

   10. Simplified 97.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+152}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.im \leq -4.5 \cdot 10^{-71}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 1.18 \cdot 10^{-125}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \mathbf{elif}\;x.im \leq 10^{+82}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \end{array} \]

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ t_1 := x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + t_0\\ t_2 := \left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{if}\;x.im \leq -8.5 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -4.5 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq 10^{-125}:\\ \;\;\;\;t_0 + \frac{x.re + x.im}{\frac{\frac{1}{x.im}}{x.re}}\\ \mathbf{elif}\;x.im \leq 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (let* ((t_0 (* x.re (+ (* x.re x.im) (* x.re x.im))))
        (t_1 (+ (* x.im (- (* x.re x.re) (* x.im x.im))) t_0))
        (t_2 (+ (+ x.im x.im) (* (* x.im (- x.re x.im)) (+ x.re x.im)))))
   (if (<= x.im -8.5e+151)
     t_2
     (if (<= x.im -4.5e-71)
       t_1
       (if (<= x.im 1e-125)
         (+ t_0 (/ (+ x.re x.im) (/ (/ 1.0 x.im) x.re)))
         (if (<= x.im 1e+81) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0;
	double t_2 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	double tmp;
	if (x_46_im <= -8.5e+151) {
		tmp = t_2;
	} else if (x_46_im <= -4.5e-71) {
		tmp = t_1;
	} else if (x_46_im <= 1e-125) {
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / x_46_re));
	} else if (x_46_im <= 1e+81) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x_46re * ((x_46re * x_46im) + (x_46re * x_46im))
    t_1 = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + t_0
    t_2 = (x_46im + x_46im) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
    if (x_46im <= (-8.5d+151)) then
        tmp = t_2
    else if (x_46im <= (-4.5d-71)) then
        tmp = t_1
    else if (x_46im <= 1d-125) then
        tmp = t_0 + ((x_46re + x_46im) / ((1.0d0 / x_46im) / x_46re))
    else if (x_46im <= 1d+81) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	double t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0;
	double t_2 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	double tmp;
	if (x_46_im <= -8.5e+151) {
		tmp = t_2;
	} else if (x_46_im <= -4.5e-71) {
		tmp = t_1;
	} else if (x_46_im <= 1e-125) {
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / x_46_re));
	} else if (x_46_im <= 1e+81) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im))
	t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0
	t_2 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))
	tmp = 0
	if x_46_im <= -8.5e+151:
		tmp = t_2
	elif x_46_im <= -4.5e-71:
		tmp = t_1
	elif x_46_im <= 1e-125:
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / x_46_re))
	elif x_46_im <= 1e+81:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im)
	t_0 = Float64(x_46_re * Float64(Float64(x_46_re * x_46_im) + Float64(x_46_re * x_46_im)))
	t_1 = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + t_0)
	t_2 = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)))
	tmp = 0.0
	if (x_46_im <= -8.5e+151)
		tmp = t_2;
	elseif (x_46_im <= -4.5e-71)
		tmp = t_1;
	elseif (x_46_im <= 1e-125)
		tmp = Float64(t_0 + Float64(Float64(x_46_re + x_46_im) / Float64(Float64(1.0 / x_46_im) / x_46_re)));
	elseif (x_46_im <= 1e+81)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	t_0 = x_46_re * ((x_46_re * x_46_im) + (x_46_re * x_46_im));
	t_1 = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + t_0;
	t_2 = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	tmp = 0.0;
	if (x_46_im <= -8.5e+151)
		tmp = t_2;
	elseif (x_46_im <= -4.5e-71)
		tmp = t_1;
	elseif (x_46_im <= 1e-125)
		tmp = t_0 + ((x_46_re + x_46_im) / ((1.0 / x_46_im) / x_46_re));
	elseif (x_46_im <= 1e+81)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := Block[{t$95$0 = N[(x$46$re * N[(N[(x$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im + x$46$im), $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]}, If[LessEqual[x$46$im, -8.5e+151], t$95$2, If[LessEqual[x$46$im, -4.5e-71], t$95$1, If[LessEqual[x$46$im, 1e-125], N[(t$95$0 + N[(N[(x$46$re + x$46$im), $MachinePrecision] / N[(N[(1.0 / x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1e+81], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -8.50000000000000051e151 or 9.99999999999999921e80 < x.im

   1. Initial program 73.6%

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

      1. +-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. fma-def 80.5%

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

      4. *-commutative 80.5%

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

      5. distribute-lft-out 80.5%

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

      6. *-commutative 80.5%

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

   3. Simplified 80.5%

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

      1. fma-udef 73.6%

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

      2. distribute-lft-in 73.6%

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

      3. flip-+ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. +-inverses 0.0%

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

      7. +-inverses 0.0%

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

      8. flip-+ 72.2%

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

      9. distribute-lft-in 72.2%

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

      10. flip-+ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. +-inverses 0.0%

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

      15. flip-+ 80.5%

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

      16. *-commutative 80.5%

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

      17. difference-of-squares 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 -8.50000000000000051e151 < x.im < -4.5000000000000002e-71 or 1.00000000000000001e-125 < x.im < 9.99999999999999921e80

   1. Initial program 99.7%

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

   if -4.5000000000000002e-71 < x.im < 1.00000000000000001e-125

   1. Initial program 79.4%

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

      1. difference-of-squares 79.4%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. flip-+ 79.3%

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

      5. associate-*l/ 74.7%

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

   3. Applied egg-rr 74.7%

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

      1. expm1-log1p-u 62.2%

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

      2. expm1-udef 49.6%

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

      3. associate-/l* 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. expm1-def 64.9%

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

      2. expm1-log1p 79.4%

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

      3. difference-of-squares 79.4%

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

      4. associate-/l* 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x.re around inf 97.8%

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

      1. associate-/r* 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8.5 \cdot 10^{+151}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{elif}\;x.im \leq -4.5 \cdot 10^{-71}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{elif}\;x.im \leq 10^{-125}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right) + \frac{x.re + x.im}{\frac{\frac{1}{x.im}}{x.re}}\\ \mathbf{elif}\;x.im \leq 10^{+81}:\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \end{array} \]

Alternative 5: 83.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -370 \lor \neg \left(x.im \leq 420\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -370.0) (not (<= x.im 420.0)))
   (+ (* x.im (- (* x.re x.re) (* x.im x.im))) (+ x.im x.im))
   (* x.re (* x.re (* x.im 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -370.0) || !(x_46_im <= 420.0)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-370.0d0)) .or. (.not. (x_46im <= 420.0d0))) then
        tmp = (x_46im * ((x_46re * x_46re) - (x_46im * x_46im))) + (x_46im + x_46im)
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -370.0) || !(x_46_im <= 420.0)) {
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -370.0) or not (x_46_im <= 420.0):
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im)
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -370.0) || !(x_46_im <= 420.0))
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re * x_46_re) - Float64(x_46_im * x_46_im))) + Float64(x_46_im + x_46_im));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -370.0) || ~((x_46_im <= 420.0)))
		tmp = (x_46_im * ((x_46_re * x_46_re) - (x_46_im * x_46_im))) + (x_46_im + x_46_im);
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -370.0], N[Not[LessEqual[x$46$im, 420.0]], $MachinePrecision]], N[(N[(x$46$im * N[(N[(x$46$re * x$46$re), $MachinePrecision] - N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im + x$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -370 or 420 < x.im

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      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-+ 77.0%

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

      8. *-commutative 77.0%

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

      9. distribute-lft-in 77.0%

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

      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-+ 85.4%

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

   3. Applied egg-rr 85.4%

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

   if -370 < x.im < 420

   1. Initial program 85.5%

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

      1. difference-of-squares 85.5%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. flip-+ 85.5%

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

      5. associate-*l/ 78.6%

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

   3. Applied egg-rr 78.6%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 49.6%

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

      3. associate-/l* 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. expm1-def 69.2%

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

      2. expm1-log1p 85.4%

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

      3. difference-of-squares 85.4%

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

      4. associate-/l* 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x.im around 0 75.8%

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

      1. distribute-lft1-in 75.8%

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

      2. metadata-eval 75.8%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. metadata-eval 75.8%

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

      6. distribute-rgt1-in 75.8%

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

      7. unpow2 75.8%

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

      8. associate-*r* 90.0%

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

      9. distribute-rgt1-in 90.0%

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

      10. metadata-eval 90.0%

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

      11. *-commutative 90.0%

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

   10. Simplified 90.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -370 \lor \neg \left(x.im \leq 420\right):\\ \;\;\;\;x.im \cdot \left(x.re \cdot x.re - x.im \cdot x.im\right) + \left(x.im + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 6: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -185 \lor \neg \left(x.im \leq 900\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im)
 :precision binary64
 (if (or (<= x.im -185.0) (not (<= x.im 900.0)))
   (+ (+ x.im x.im) (* (* x.im (- x.re x.im)) (+ x.re x.im)))
   (* x.re (* x.re (* x.im 3.0)))))

C

double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -185.0) || !(x_46_im <= 900.0)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8) :: tmp
    if ((x_46im <= (-185.0d0)) .or. (.not. (x_46im <= 900.0d0))) then
        tmp = (x_46im + x_46im) + ((x_46im * (x_46re - x_46im)) * (x_46re + x_46im))
    else
        tmp = x_46re * (x_46re * (x_46im * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im) {
	double tmp;
	if ((x_46_im <= -185.0) || !(x_46_im <= 900.0)) {
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	} else {
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im):
	tmp = 0
	if (x_46_im <= -185.0) or not (x_46_im <= 900.0):
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im))
	else:
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0))
	return tmp

Julia

function code(x_46_re, x_46_im)
	tmp = 0.0
	if ((x_46_im <= -185.0) || !(x_46_im <= 900.0))
		tmp = Float64(Float64(x_46_im + x_46_im) + Float64(Float64(x_46_im * Float64(x_46_re - x_46_im)) * Float64(x_46_re + x_46_im)));
	else
		tmp = Float64(x_46_re * Float64(x_46_re * Float64(x_46_im * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im)
	tmp = 0.0;
	if ((x_46_im <= -185.0) || ~((x_46_im <= 900.0)))
		tmp = (x_46_im + x_46_im) + ((x_46_im * (x_46_re - x_46_im)) * (x_46_re + x_46_im));
	else
		tmp = x_46_re * (x_46_re * (x_46_im * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_] := If[Or[LessEqual[x$46$im, -185.0], N[Not[LessEqual[x$46$im, 900.0]], $MachinePrecision]], N[(N[(x$46$im + x$46$im), $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], N[(x$46$re * N[(x$46$re * N[(x$46$im * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -185 or 900 < x.im

   1. Initial program 84.0%

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

      1. +-commutative 84.0%

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

      2. *-commutative 84.0%

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

      3. fma-def 88.2%

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

      4. *-commutative 88.2%

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

      5. distribute-lft-out 88.2%

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

      6. *-commutative 88.2%

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

   3. Simplified 88.2%

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

      1. fma-udef 84.0%

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

      2. distribute-lft-in 84.0%

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

      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-+ 77.0%

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

      9. distribute-lft-in 77.0%

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

      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-+ 85.4%

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

      16. *-commutative 85.4%

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

      17. difference-of-squares 97.1%

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

      18. associate-*r* 97.1%

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

      19. *-commutative 97.1%

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

   5. Applied egg-rr 97.1%

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

   if -185 < x.im < 900

   1. Initial program 85.5%

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

      1. difference-of-squares 85.5%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

      4. flip-+ 85.5%

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

      5. associate-*l/ 78.6%

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

   3. Applied egg-rr 78.6%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 49.6%

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

      3. associate-/l* 49.6%

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

   5. Applied egg-rr 49.6%

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

      1. expm1-def 69.2%

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

      2. expm1-log1p 85.4%

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

      3. difference-of-squares 85.4%

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

      4. associate-/l* 99.7%

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

      5. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x.im around 0 75.8%

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

      1. distribute-lft1-in 75.8%

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

      2. metadata-eval 75.8%

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

      3. associate-*r* 75.8%

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

      4. *-commutative 75.8%

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

      5. metadata-eval 75.8%

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

      6. distribute-rgt1-in 75.8%

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

      7. unpow2 75.8%

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

      8. associate-*r* 90.0%

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

      9. distribute-rgt1-in 90.0%

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

      10. metadata-eval 90.0%

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

      11. *-commutative 90.0%

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

   10. Simplified 90.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -185 \lor \neg \left(x.im \leq 900\right):\\ \;\;\;\;\left(x.im + x.im\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \left(x.re \cdot \left(x.im \cdot 3\right)\right)\\ \end{array} \]

Alternative 7: 50.2% 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 84.8%

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

   1. +-commutative 84.8%

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

   2. *-commutative 84.8%

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

   3. distribute-lft-out 84.8%

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

   4. associate-*l* 84.8%

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

   5. *-commutative 84.8%

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

   6. distribute-lft-out 86.8%

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

   7. associate-+r- 86.8%

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

   8. distribute-lft-out-- 79.7%

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

3. Simplified 79.8%

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

   1. sub-neg 79.8%

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

   2. associate-*l* 79.8%

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

   3. associate-*l* 87.4%

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

5. Applied egg-rr 87.4%

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

6. Taylor expanded in x.re around 0 87.4%

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

7. Taylor expanded in x.re around inf 52.3%

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

   1. unpow2 52.3%

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

9. Simplified 52.3%

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

10. Final simplification 52.3%

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

Alternative 8: 56.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. difference-of-squares 85.6%

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

   2. associate-*r* 93.1%

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

   3. *-commutative 93.1%

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

   4. flip-+ 84.8%

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

   5. associate-*l/ 77.5%

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

3. Applied egg-rr 77.5%

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

   1. expm1-log1p-u 53.9%

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

   2. expm1-udef 45.7%

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

   3. associate-/l* 47.0%

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

5. Applied egg-rr 47.0%

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

   1. expm1-def 57.4%

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

   2. expm1-log1p 84.7%

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

   3. difference-of-squares 85.5%

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

   4. associate-/l* 93.1%

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

   5. +-commutative 93.1%

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

7. Simplified 93.1%

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

8. Taylor expanded in x.im around 0 52.3%

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

   1. distribute-lft1-in 52.3%

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

   2. metadata-eval 52.3%

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

   3. associate-*r* 52.3%

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

   4. *-commutative 52.3%

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

   5. metadata-eval 52.3%

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

   6. distribute-rgt1-in 52.3%

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

   7. unpow2 52.3%

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

   8. associate-*r* 59.9%

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

   9. distribute-rgt1-in 59.9%

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

   10. metadata-eval 59.9%

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

   11. *-commutative 59.9%

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

10. Simplified 59.9%

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

11. Final simplification 59.9%

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

Alternative 9: 34.8% 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 84.8%

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

   1. *-commutative 84.8%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. flip-+ 64.3%

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

   8. *-commutative 64.3%

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

   9. distribute-lft-in 64.3%

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

   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-+ 53.1%

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

3. Applied egg-rr 53.1%

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

4. Taylor expanded in x.re around inf 39.3%

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

   1. unpow2 39.3%

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

6. Simplified 39.3%

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

7. Final simplification 39.3%

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

Alternative 10: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -3.25 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 -3.25)

C

double code(double x_46_re, double x_46_im) {
	return -3.25;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = -3.25d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return -3.25;
}

Python

def code(x_46_re, x_46_im):
	return -3.25

Julia

function code(x_46_re, x_46_im)
	return -3.25
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = -3.25;
end

Wolfram

code[x$46$re_, x$46$im_] := -3.25

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. *-commutative 84.8%

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

   3. distribute-lft-out 84.8%

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

   4. associate-*l* 84.8%

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

   5. *-commutative 84.8%

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

   6. distribute-lft-out 86.8%

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

   7. associate-+r- 86.8%

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

   8. distribute-lft-out-- 79.7%

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

3. Simplified 79.8%

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

   1. flip3-- 12.3%

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

   2. frac-2neg 12.3%

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

   3. *-commutative 12.3%

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

   4. unpow-prod-down 12.2%

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

   5. metadata-eval 12.2%

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

   6. associate-*l* 12.2%

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

   7. pow-pow 12.2%

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

   8. metadata-eval 12.2%

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

5. Applied egg-rr 8.7%

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

7. Simplified 2.6%

   \[\leadsto \color{blue}{-3.25} \]

8. Final simplification 2.6%

   \[\leadsto -3.25 \]

Alternative 11: 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 84.8%

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

2. Taylor expanded in x.re around 0 68.0%

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

4. Simplified 24.3%

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

   1. *-commutative 24.3%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. flip-+ 7.8%

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

   8. *-commutative 7.8%

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

   9. distribute-rgt-in 7.8%

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

   10. *-commutative 7.8%

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

   11. flip-+ 0.0%

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

   12. clear-num 0.0%

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

   13. *-commutative 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. *-commutative 0.0%

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

   17. *-commutative 0.0%

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

   18. +-inverses 0.0%

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

   19. +-inverses 0.0%

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

   20. flip-+ 3.0%

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

6. Applied egg-rr 3.0%

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

7. Taylor expanded in x.im around inf 2.6%

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

8. Final simplification 2.6%

   \[\leadsto -3 \]

Alternative 12: 2.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 3.25 \end{array} \]

FPCore

(FPCore (x.re x.im) :precision binary64 3.25)

C

double code(double x_46_re, double x_46_im) {
	return 3.25;
}

Fortran

real(8) function code(x_46re, x_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    code = 3.25d0
end function

Java

public static double code(double x_46_re, double x_46_im) {
	return 3.25;
}

Python

def code(x_46_re, x_46_im):
	return 3.25

Julia

function code(x_46_re, x_46_im)
	return 3.25
end

MATLAB

function tmp = code(x_46_re, x_46_im)
	tmp = 3.25;
end

Wolfram

code[x$46$re_, x$46$im_] := 3.25

Derivation

1. Initial program 84.8%

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

   1. +-commutative 84.8%

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

   2. *-commutative 84.8%

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

   3. distribute-lft-out 84.8%

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

   4. associate-*l* 84.8%

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

   5. *-commutative 84.8%

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

   6. distribute-lft-out 86.8%

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

   7. associate-+r- 86.8%

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

   8. distribute-lft-out-- 79.7%

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

3. Simplified 79.8%

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

   1. flip3-- 12.3%

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

   2. div-inv 12.2%

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

   3. *-commutative 12.2%

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

   4. unpow-prod-down 12.2%

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

   5. metadata-eval 12.2%

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

   6. associate-*l* 12.2%

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

   7. pow-pow 12.2%

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

   8. metadata-eval 12.2%

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

   9. associate-+r+ 12.2%

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

5. Applied egg-rr 8.7%

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

7. Simplified 2.8%

   \[\leadsto \color{blue}{3.25} \]

8. Final simplification 2.8%

   \[\leadsto 3.25 \]

Developer target: 92.1% 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 2023283 
(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)))